If your life's work can be accomplished in your lifetime, you're not thinking big enough. Wes Jacks

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The importance of being hierarchical

Knock, And He'll open the door. Vanish, And He'll make you shine like the sun. Fall, And He'll raise

Being Seen Being Counted

If you feel beautiful, then you are. Even if you don't, you still are. Terri Guillemets

The Price of Friendship

How wonderful it is that nobody need wait a single moment before starting to improve the world. Anne

the price of justice

In the end only three things matter: how much you loved, how gently you lived, and how gracefully you

The Price of Experience

Open your mouth only if what you are going to say is more beautiful than the silience. BUDDHA

The Price of Variance Risk

Come let us be friends for once. Let us make life easy on us. Let us be loved ones and lovers. The earth

The Notion of Price Homogeneity

In the end only three things matter: how much you loved, how gently you lived, and how gracefully you

The Price of Street Friends

Seek knowledge from cradle to the grave. Prophet Muhammad (Peace be upon him)

The Price of Being Near-Sighted Fabian Kuhn, Thomas Moscibroda, Roger Wattenhofer {kuhn,moscitho,wattenhofer}@tik.ee.ethz.ch Computer Engineering and Networks Laboratory, ETH Zurich, 8092 Zurich, Switzerland

Abstract Achieving a global goal based on local information is challenging, especially in complex and large-scale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible tradeoff between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρ∆)1/k -approximation for general covering and packing problems in time O(k2 ), where ρ depends on the LP’s coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n1/k ) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its k-neighborhood.

1 Introduction Many of the most fascinating and fundamental systems in the world are large and complex networks, such as the human society, the Internet, or the human brain. Such systems have in common that their entirety is composed of a multiplicity of individual entities; human beings in society, hosts in the Internet, or neurons in the brain. As diverse as these systems may be, they share the key characteristic that the capability of direct communication of each individual entity is restricted to only a small subset of neighboring entities. Most human communication, for instance, is between acquaintances or within the family, and neurons are directly linked with merely a relatively small number of other neurons for neurotransmission. On the other hand, in spite of each node thus being inherently “near-sighted,” i.e., restricted to local communication, the entirety of the system is supposed to come up with some kind of global solution, or to keep an equilibrium. Achieving a global goal based on local information is challenging. Many of the systems which are the focus of computer science fall exactly into the above mentioned category of networks. In the Internet, largescale peer-to-peer systems, or mobile ad hoc and sensor networks, no node in the network is capable of keeping

global information on the network. Instead, these nodes have to perform their intended (global) task based on local information only. In other words, all computation in these systems is inherently local! Not surprisingly, studying the fundamental possibilities and limitations of local computation is therefore of interest to theoreticians in approximation theory, distributed computing, and graph theory. The study of local computation has been initiated by the pioneering work of Linial [11], and Naor and Stockmeyer [15] more than a decade ago. Also, the work of Peleg [17] has resulted in numerous interesting and deep results. But nonetheless, there remains a great number of important open problems related to questions such as what kind of global tasks can be performed by individual entities that have to base their decisions on local information, or how much local information is required in order to come up with a globally optimal solution. For instance, the open question from [16], that is, characterizing the trade-off between communication among agents to exchange information and the global utility achieved, has been unanswered. It is the goal of this paper to make a step towards answering this open problem and, more generally, to bring forward some of the underlying principles and trade-offs governing local computation. Not surprisingly, many global criteria such as counting the total number of nodes in the network or obtaining a minimum spanning tree cannot be met if every node’s decision is based solely on local knowledge. On the other hand, many fundamental coordination tasks and applications in large-scale networks appear to be easier to handle from a “local-global” perspective. Specifically, classic graph theory problems such as dominating set or matching can be formulated as standard covering and packing problems. The nature of simple covering and packing problems like minimum vertex cover or maximum matching appears to be local and intuitively, one may think that each node’s (edge’s) decision is not affected by very distant nodes (edges). Interestingly, we prove in this paper that this intuition is misleading even for the most basic packing problems.

On the positive side, we show that there exist distributed approximation algorithms that almost achieve the optimal trade-off, even in the practically important case in which the amount of information exchanged in each message is limited. Specifically, we give the following results: • Consider a network with n nodes and maximum degree ∆. Assume that each node in a network graph has to base its decision on its k-hop neighborhood. We present an efficient deterministic distributed algorithm that operates with small messages of size O(log n) bits. The algorithm achieves a (ρ∆)1/k -approximation for general covering and packing problems in O(k 2 ) communication rounds, where ρ depends on the coefficients of the underlying LP. • When message size is unbounded, each network node can easily gather the entire information from its O(k)-neighborhood in O(k) communication rounds. Hence, in this case, the achievable trade-off is a true consequence of locality restriction only. We present an algorithm producing an O(n1/k )-approximation if each node knows its O(k)-neighborhood. • In combination with (distributed) randomized rounding, the above algorithms can be transformed into constant-time distributed algorithms having non-trivial approximation ratios for various combinatorial problems.

message model, respectively. The packing lower bound is derived in subsequent Section 6. Section 7 concludes the paper. Due to lack of space, some proofs are omitted from this extended abstract.1 2 Related Work Little is known about the fundamental limitations of locality-based approaches. Fich and Ruppert, for instance, describe a numerous lower bounds and impossibility results in distributed computing [5]. But most of them apply to other computational models where locality is no issue or there are additional, more restrictive limiting factors, such as bounded message size [4]. There have been virtually no nontrivial lower bounds for local computation, besides Linial’s seminal Ω(log∗ n) time lower bound for constructing a maximal independent set on a ring [11]. In addition, we have shown that minimum vertex cover and thus covering prob2 lems cannot be approximated better than Ω(nc/k /k) and Ω(∆1/k/k) if each node’s information is restricted to its k-neighborhood [9]. On the positive side, it was shown by Naor and Stockmeyer [15] that there exist locally checkable labelings which can be computed in distributed constant time, i.e., with completely local information only. The focus of this paper is to understand locality in problems that can be formulated as packing and covering LPs. There are a number of (parallel) algorithms for solving such LPs which are faster than interior-point methods that can be applied to general LPs (e.g. [6, 14, 18, 22]). All these algorithms need at least some global information to work. The problem of approximating positive LPs using only local information has been introduced in [16]. The first algorithm achieving a constant approximation in polylogarithmic time is described in [2]. Distributed algorithms targeted for specific covering and packing problems include algorithms for the minimum dominating set problem [3, 8, 19] as well as algorithms for maximal matchings and maximal independent sets [1, 7, 13]. This implies a constant approximation for maximum matching. All described distributed algorithms have a time complexity which is at least logarithmic in n. That is, each node may gather information which is as far away as O(log n) hops. Hence, while these algorithms provide solutions to particular problems, they do not fully explore the trade-off between local knowledge and solution quality. A distributed algorithm for minimum dominating set running in an arbitrary, possibly constant number of rounds is found in [10].

• Finally, we show that the trade-off achieved by our algorithms is almost tight. Specifically, we prove that even the most simple packing problem, (fractional) maximum matching, cannot be ap2 1/k proximated within Ω(nc/k /k) p and Ω(∆ /k), respectively. This implies Ω( log n/ log log n) and Ω(log ∆/ log log ∆) time lower bounds for (possibly randomized) distributed algorithms in order to obtain a constant or polylogarithmic approximation for maximum matching and packing problems, even if message size is unbounded. This lower bound extends a similar result that has recently been achieved for the minimum vertex cover problem in [9]. Note that by giving upper and lower bounds for general covering and packing LPs, we show that many different natural problems behave similarly with regard to local approximability. This is of great theoretical interest since such a classification of problems may provide a completely new insight into the impact of locality on algorithms. Related work and the model of computation are described in Sections 2 and 3. In subsequent Sections 4 1 A version containing all proofs can be found as TIK techniand 5, we give distributed algorithm for general covercal report 229 at ftp://ftp.tik.ee.ethz.ch/pub/publications/TIKing and packing LPs in the bounded and unbounded Report229.pdf

3 Model We describe the network as an undirected graph G = (V, E). The vertices V = {v1 , . . . , vn } represent the network entities or nodes (e.g. processors) and the edges represent bidirectional communication channels. We distinguish two prototypical and classic message passing models [17], LOCAL and CON GEST , depending on how much information can be sent in each message. In the LOCAL model (e.g., [11, 15, 17]), knowing your k-neighborhood and performing k communication rounds are equivalent. It is assumed that in every communication round, each node in the network can send an arbitrarily long message to each of its neighbors. Local computations are for free. Each node has a unique identifier and initially, nodes have no knowledge about the network graph. In k communication rounds, a node v may collect the IDs and interconnections of all nodes up to distance k from v, because messages are unbounded. Hence, each node has a partial (local) view of the graph; it knows its entire vicinity up to distance k. Let Tv,k be the topology seen by v after k rounds. Tv,k is the graph induced by the k-neighborhood of v without all edges between nodes at distance exactly k. The labeling (i.e., the assignment of IDs) of Tv,k is denoted by L(Tv,k ). The view of a node v is the pair, Vv,k := (Tv,k , L(Tv,k )). The view of an edge e = (u, v) is the union of views of its incident nodes. The best a local algorithm can do in time k, is to have every node v collect its k-neighborhood and base its decision on Vv,k . Since the LOCAL model abstracts away other aspects arising in the design of distributed algorithms (congestion, fast local computation, . . . ), it is the most fundamental model when studying the phenomenon of locality; particularly for lower bounds. In practice, the amount of information exchanged between two neighbors in one communication step is limited. The CON GEST model [4, 17] takes into account the volume of communication. This model limits the information that can be sent in one message to O(log n) bits. Given this additional restriction, even problems on the complete network graph, which could be solved optimally in a single communication round in the LOCAL model, become nontrivial. A fractional covering problem (PP) and its dual fractional packing problem (DP), are linear programs of the canonical forms min s.t.

cT x A·x≥b x≥0

max s.t.

bT y AT · y ≤ c y≥0

where all aij , bi , and ci are non-negative. We will use

the term primal LP (PP) for the minimization and dual LP (DP) for the maximization problem. The number of primal and dual variables are denoted by m and n, respectively. Let amax := maxi,j {aij , bi , ci } be the maximum coefficient and amin := mini,j {aij , bi , ci } \ {0} be the minimum non-zero coefficient of (PP) and (DP). ρ := amax /amin is the maximum ratio between any two coefficients. Analogously to [2, 16], we consider the following distributed setting. The linear program is bound to a network graph G = (V, E). Each primal variable xi and (p) each dual variable yj is associated with a node vi ∈ V (d) and vj ∈ V , respectively. There are communication links between primal and dual nodes wherever the respective variables occur in the corresponding inequality. (p) (d) Thus, (vi , vj ) ∈ E if and only if xi occurs in the j th (p)

inequality of (PP), i.e., vi 2

aji > 0 . The degrees of (d) δj ,

(d)

and vj

are connected iff

(p) vi

(d) (p) and vj are called δi (p) maxi δi and ∆d := maxj

and (d)

respectively. ∆p := δj are called the primal and dual degree, respectively. The (p) (p) set of dual neighbors of vi is denoted by Ni , the set (d) (d) of primal neighbors of vi by Ni . Where convenient, (p) (d) Ni and Nj also denote the sets of the indices of the respective nodes. 4 Bounded Messages In this section, we describe an efficient distributed algorithm to approximate covering and packing linear programs in the CON GEST model. For our algorithm, we need the LPs (PP) and (DP) to be of the following special form: (4.1)

∀i, j : bi = 1,

aij = 0 or aij ≥ 1.

The transformation to (4.1) is done in two steps. First, every aij is replaced by a ˆij := aij /bi and bi is replaced by 1. In the second step, the ci and a ˆij are divided by λi := minj {ˆ aji } \ {0}. The optimal objective values of the transformed LPs are the same. A feasible solution for the transformed LP (4.1) can be converted to a feasible solution of the original LP by dividing all x-values by the corresponding λi and by dividing the y-values by the corresponding bi . This conserves the values of the objective functions. Note that the described transformation can be computed locally in a constant number of rounds. For the rest of this section, 2 Note that in order to solve such a problem in a real network setting where only some variables correspond to nodes, the other variables may be simulated by the nodes as well. Variables associated to edges (like in vertex cover or maximum matching) can be simulated by incident nodes.

LP Approximation (p) Algorithm for Primal Node vi :

LP Approximation (d) Algorithm for Dual Node vi : yi := yi+ := wi := fi := 0; ri := 1; for ep := kp − 2 to −f − 1 by −1 do for 1 to h do r˜i := ri ; 5: for ed := kd − 1 to 0 by −1 do

xi := 0; for ep := kp − 2 to −f − 1 by −1 do for 1 to h do P (∗ γi := cmax j aji rj ∗) ci 5: for ed := kd − 1 to 0 by −1 do P a r ˜ 6: γ˜i := cmax j ji j ; ci 1: 2: 3: 4:

7: 8: 9: 10:

if

e /k γ˜i ≥ Γpp p then ed /kd x+ ; xi i := 1/Γd

1: 2: 3: 4: 6: 7:

x+ i ;

:= xi + fi; send x+ i , γ˜i to dual neighbors;

8: 9: 10:

11:

11:

12:

12:

13: 14: 15: 16: 17:

13: 14: 15: 16: 17:

receive r˜j from dual neighbors od;

receive rj from dual neighbors od od; P 21: xi := xi / minj∈N (p) ` aj` x` 18: 19: 20:

18: 19: 20: 21:

i

receive x+ j , γ˜jPfrom primal neighbors; yi+ := yi+ + r˜i j aij x+ j /γ˜j ; P wi+ := j aij x+ ; j wi := wi + wi+ ; fi := fi + wi+ ; if wi ≥ 1 then r˜i := 0 fi; send r˜i to primal neighbors od; increase duals(); send ri to primal neighbors od od; P yi := yi / maxj∈N (d) c1j ` a`j y` i

Algorithm 1: Distributed LP Approximation Algorithm

we assume that the coefficients of the LP are given according to (4.1). We start the description of the algorithm with a general outline. As our algorithm borrows from the greedy dominating set/set cover algorithm, it is useful to view the distributed LP algorithm in this context. The greedy minimum dominating set (MDS) algorithm starts with an empty set and sequentially adds the node which covers the most not yet covered nodes. The LP relaxation of MDS asks for variables xi for the nodes vi such that the sum of the xi in the 1-neighborhood of every node is at least 1. Analogous to the sequential greedy approach, we also start with all xi set to 0 and we give priority to nodes with many uncovered neighbors when increasing the xi . In particular, we always increase the xi of all the nodes whose number of uncovered neighbors is maximum up to a certain factor (active nodes). In order not to ‘over-cover’ a node with many active neighbors, we have to carefully choose the increment of the xi at active nodes. As we proceed, we simultaneously compute a solution for the dual LP such that the objective values of the solutions stay the same. In the end, each node is covered at least f times and each dual constraint is fulfilled up to a factor αf . Hence by dividing by f and αf , we obtain feasible, α-

approximate primal and dual solutions, respectively. In order to achieve that every primal inequality is (d) fulfilled f times, each dual node vi needs a requirement ri ≤ 1 which is decreased every time the corresponding primal constraint is achieved and a variable fi which counts how many times the primal constraint has been fulfilled (cf. [21]). The decision whether a primal node (p) vi is active and can increase xi is based on the efficiency per cost ratio γi which is defined as follows: γi :=

cmax X aji rj . ci j

For simplicity, we assume that all nodes know cmax := max{ci } as well as two other global quantities Γp and Γd which are defined as Γp := max i

n cmax X · aji ci j=1

and Γd := max i

m X

aij .

j=1

At the price of a considerably more complicated (and less readable) algorithm, it is possible to get rid of this assumption. For details, we refer to the full paper. The detailed algorithm is given by Algorithm 1 along with the procedure increase duals() which is

(p)

Lemma 4.3. Let vi be a primal node and let Yi := P a y j ji j be the weighted sum of the y-values of its dual neighbors. Further, let Yi+ be the increase of Yi and γi− be the decrease of γi during an execution of increase duals(). We have

procedure increase duals(): 1: if wi ≥ 1 then 2: if fi ≥ f then 3: yi := yi + yi+ ; yi+ := 0; 4: ri := 0; wi := 0 5: else if wi ≥ 2 then 6: yi := yi + yi+ ; yi+ := 0; bw c/k 7: ri := ri /Γp i p 8: else 1/k 1/k 9: λ := max{Γd d , Γp p }; e /k 10: yi := yi + min{yi+ , ri λ/Γpp p }; e /k 11: yi+ := yi+ − min{yi+ , ri λ/Γpp p }; 1/k 12: ri := ri /Γp p 13: fi; 14: wi := wi − bwi c 15: fi

3/kp

Yi+ ≤

Γp

1/kp

· max{Γp 1/k γi (Γp p

1/kd

, Γd

}

− 1)

·

ci cmax

· γi− .

Proof. We prove the lemma by showing that the in(p) (d) equality holds for every dual neighbor vj of vi . Let βj be the increase of yj and let rj− be the decrease of rj . We show that 1/kp

(4.3)

βj ≤

1/kp

Γp

1/kd

· max{Γp

, Γd

e /k 1/k Γpp p (Γp p

− 1)

}

· rj− . (e +2)/kp

used by the dual nodes. The algorithm has two parameters kp ≥ 1 and kd ≥ 1 which determine the tradeoff between time complexity and approximation quality. The bigger kp and kd , the better the approximation ratio of the algorithm. On the other hand, smaller kp and kd lead to a faster algorithm. Algorithm 1 also makes use of two values f and h which are defined as follows: & ' & ' kp + 1 kp f := and h := 1 + 1/k . 1/k Γp p − 1 Γp p ln Γp In the following, we present lemmas which establish all the necessary details to analyze Algorithm 1. The goal of the outer ep -loop is to reduce the maximum “weighted primal degree” γi . This is reflected by the following lemma. (p)

The lemma then follows because γi ≤ Γp p (Lemma 4.1) and because X cmax X Yi+ = aji βj and γi− = aji rj− . c i j j

To prove Inequality (4.3), we again consider the cases where wj ≥ 2 and where 1 ≤ wj < 2. If wj ≥ 2, e /k 1/k by Lemma 4.2, βj = yj+ ≤ rj (1 + Γd d )/Γpp p . The 2/k

requirement rj is divided by at least Γp p and therefore 2/k 2/k rj− ≥ rj (Γp p − 1)/Γp p . Together, we get 2/kp

1/kd

βj

≤

1 + Γd ³

≤

·

e /k Γpp p

1+

1/k Γp p

e /kp

2/k Γp p

´

³

Γpp

Γp

−1

· rj−

1/k Γp p 1/kp

Γp

1/k

1/k

max{Γp p , Γd d } ´³ ´ rj− . 1/kp + 1 Γp −1

Lemma 4.1. For each primal node vi , at all times (e +2)/kp during Algorithm 1, γi ≤ Γp p .

For 1 ≤ wj < 2, the proof is along the same lines. Here, 1/k e /k 1/k 1/k βj ≤ rj max{Γp p , Γd d }/Γpp p and rj− = rj (Γp p −

One complete run (kd iterations) of the innermost ed -loop can be seen as one parallel greedy step. Primal nodes with large γi increase their xi such that the corresponding increases yi+ of the dual variables are almost feasible.

1)/Γp

Lemma 4.2. Each time a dual node enters increase duals() in Algorithm 1, (4.2)

yi+ ≤ ri ·

wi e /kp

Γpp

and yi+ ≤ ri ·

1/kd

Γd

+1

e /kp

Γpp

1/kp

. Again, we obtain Inequality (4.3): 1/kp

βj ≤

max{Γp

e /kp

Γpp

1/kd

, Γd

}

1/kp

·

Γp

1/kp

Γp

−1

· rj− .

We do not have to consider the case fj ≥ f explicitly because the same analysis as for wj ≥ 2 applies in this case. (p)

.

As shown in Lemma 4.4, all the increases of the dual variables together render the dual constraints feasible up to a small factor times (kp + f + 1). We first need the following helper lemma.

Lemma 4.4. Let vi be a primal node and Yi = P j aji yj be the weighted sum of the y-values of the dual (p)

neighbors of vi . After the main part of the algorithm (i.e., after the loops at line 20), n o ci 1/kd 1/kp p (kp + f + 1)Γ3/k max Γ , Γ . Yi ≤ p p d cmax

Proof. For simplicity, we define Q :=

1 3/kp 1/k p Γ max{Γ1/k , Γd d }. p cmax p

Before γi is decreased for the last time, we have γi ≥ (f −1)/kp 1/Γp because at least one rj in the dual neigh(p) borhood of vi has to be greater than 0. If we assume that the last time γi is decreased it is only reduced to (f +1)/kp γi = 1/Γp , Lemma 4.3 still holds. The analysis is exactly the same as for the case wj ≥ 2 in Lemma 4.3. By Lemma 4.3, Yi is therefore bounded by the 1/k area under the curve ci Q/(Γp p − 1) · 1/x for x between (f +1)/kp 1/Γp and Γp : Z Γp ci Q 1 · dx Yi ≤ 1/k p 1 x Γp −1 (f +1)/k

Because fj ≥ f , all yj+ are 0 in the end and thus yj is equal to the sum of all increases of yj+ . ¤ Combining the above lemmas, we get the following theorem. Theorem 4.1. For arbitrary kp , kd ≥ 1, Algorithm 1 approximates (PP) and (DP) by a factor n o 1/k p p Γ4/k max Γ1/k , Γd d . p p The time complexity of Algorithm 1 is Ã Ã !Ã !! 1 kp O kd kp 1 + 1/k 1 + 1/k . Γp p − 1 Γp p log Γp For kp ∈ O(log Γp ), this simplifies to O(kd kp ).

p

Proof. For the approximation ratio, we have to look at line 21 of Algorithm 1 where all x and y values = ≤ ci (kp + f + 1)Q. are divided by the largest possible values to keep/make 1/k Γp p − 1 the primal/dual solution feasible. By Lemma 4.5, each primal constraint is satisfied at least f times. Therefore, The last inequality follows from ln(1 + t) ≤ t. ¤ all primal variables are divided by at least f . Due to At the end of the algorithm, all primal constraints Lemma 4.4, for each primal node, the sum of the y are satisfied at least f times. Further, the primal and values of its dual neighbors is at most ci (kp +f +1)Q for Q as defined in Lemma 4.4. Dividing all dual variables dual objective functions are the same. by (kp + f + 1)Q therefore renders the dual solution Lemma 4.5.PAfter the loops P at line 20, ∀i: ri = 0 and feasible. By Lemma 4.5, the ratio between the objective m n functions of the primal and the dual solutions is fi ≥ f and i=1 ci xi = cmax j=1 yj . Pm kp + f + 1 Proof. When entering the ep -loop for the last time, by Pi=1 ci xi ≤ c Q max n f y Lemma 4.1, j=1 i X X kp +1 +1 kp + 1/k Γp(−f +1)/kp ≥ γj ≥ aij ri ≥ ri . Γp p −1 ≤ c Q max (p) i kp +1 Γp

³ ´ 1/k ci (kp + f + 1)Q ln Γp p

i∈Nj

γj can only be greater than 0 if there is exactly one ri (p) in the dual neighborhood of vj which is greater than zero. If ri is still greater than 0 when ed = 0, xj will be increased by 1 which makes wj ≥ 1 and therefore ri = 0 after the next call to increase duals(). fi counts the number of times the ith constraint of (PP) is satisfied. It is increased together with wi in line 13 of Algorithm 1. Every time the integer part of wi bw c/k is increased, ri is divided by Γp i p and wi is set to wi − bwi c. Therefore, ri = 0 implies fi ≥ f . (p) Let vi be a primal node which increases xi by x+ i (d) (p) (line 8). All dual neighbors vj of vi increase yj+ by + aji r˜j x+ i /γ˜i . Hence, the sum of the yj -increases over all (p)

dual neighbors of vi

is

P X ci + x+ j aji r˜j + i = x . aji r˜j = xi cmax P γ˜i j cmax i j aji r˜j ci

1/kp

Γp

=

p Q cmax Γ1/k p

−1

o n 1/kd 1/kp p max Γ , Γ . = Γ4/k p p d

Because of the duality theorem for linear programming, this ratio is an upper bound on the approximation ratio for (PP) and (DP). As for the time complexity, note that each iteration of the inner-most loop (ed -loop) requires two rounds. Hence, the algorithm has time complexity O(kd (kp + f )h). The claim follows from substituting the actual values for f and h. For kp ∈ O(log Γp ), 1/k Γp p −1 is a constant and therefore the time complexity simplifies to O(kd kp ). ¤ Corollary 4.1. For sufficiently small ε, Algorithm 1 computes a (1 + ε)-approximation for (PP) and (DP) ¡ ¢ in O log Γp log Γd /ε4 rounds. In particular, a constant factor approximation can be achieved in time O(log Γp log Γd ).

Remark: Using methods similar to the ones described in [2, 14], it is possible to get rid of the dependency on the coefficients ρ := amax /amin . As a result, the running time and approximation ratio would depend on the number of nodes m and n instead of the degrees ∆p and ∆d . Distributed Randomized Rounding We can apply our distributed LP approximation algorithms together with standard distributed randomized rounding techniques to obtain distributed approximation algorithms for a number of combinatorial problems. We can prove that given an α-approximate solution for the LP relaxation of problems for which the matrix elements aij ∈ {0, 1}, we can compute in a constant number of rounds a O(α log ∆p )-approximation for the corresponding covering IP and a O(α∆d )-approximation for the packing IP.

For the decomposition of (PP) and (DP), we need the following lemma. 0 Lemma 5.1. Let {y10 , . . . , ym 0 } be a subset of the dual 0 variables of DP and let x1 , . . . , x0n0 be the primal variables which are adjacent to the given subset of the dual variables. Further let P P 0 and DP 0 be LPs where the matrix A0 consists only of the columns and rows corresponding to the variables in x0 and y 0 . Every feasible solution for P P 0 makes the corresponding primal inequalities in P P feasible and every feasible solution for DP 0 is feasible for DP (variables not occurring in P P 0 and DP 0 are set to 0). Further, the values of the objective functions for the optimal solutions of P P 0 and DP 0 are upper bounded by the optimal values for P P and DP .

We call P P 0 and DP 0 the sub-LPs induced by the subset 0 We apply the graph {y10 , . . . , ym 0 } of dual variables. decomposition algorithm of [12] to obtain P P 0 and DP 0 5 Unbounded Messages (as in Lemma 5.1) which can be solved locally. In [12], Linial and Saks presented a randomized disFor the decomposition of the linear program, we tributed algorithm to decompose a graph into sub- define G such that the node set V is the set of dual graphs of limited diameter. We use their algorithm to nodes of the graph G and the edge set E is decompose the linear program into sub-programs which ¯ © ª can be solved locally in the LOCAL model. For a genE := (u, v) ¯ u, v ∈ V ∧ dG (u, v) ≤ 4 . eral graph G = (V, E) with n nodes, the algorithm of [12] yields a subset S ⊆ V of V such that each node By this, we can guarantee that non-adjacent nodes in u ∈ S has a leader `(u) ∈ V and such that the following G do not have neighboring primal nodes in G whose variables occur in the same constraint of (PP). Further, properties hold.3 a message over an edge of G can be sent in 4 rounds (I) ∀u ∈ S : d(u, `(u)) < k on the network graph G. The basic algorithm for a (II) ∀u, v ∈ S : `(u) 6= `(v) −→ (u, v) 6∈ E. dual node v to approximate P P and DP then works as (III) S can be computed in k rounds. follows: (IV) ∀u ∈ V : Pr[u ∈ S] ≥ en11/k . 1: Run graph decomposition of [12] on G; 2: if v ∈ S then d(u, v) denotes the distance between two nodes u and 3: send IDs of primal neighbors to `(v). v on G. We apply the algorithm of [12] to obtain con4: fi; nected components of G with the following properties. 5: if v = `(u) for some u ∈ S then (I) The components have small diameter. 6: compute local PLP/DLP (cf. Lemma 5.1) (II) Different components are far enough from each of variables of u ∈ S for which v = `(u). other such that we can define a local linear program 7: send resulting values to nodes holding the for each component in a way in which the LPs of respective variables. any two components do not interfere. 8: fi (III) Each node belongs to one of the components with The primal nodes only forward messages in steps 1, probability at least p, where p depends on the 3, and 7 and receive the values for their variables in diameter we allow the components to have. step 7. We now have a closer look at the locally Because of the limited diameter, the LPs of each com- computed LPs in line 6. By Property (II) of the graph ponent can then be computed locally. We apply the de- decomposition algorithm, dual variables belonging to composition process in parallel often enough such that different local LPs cannot occur in the same dual w.h.p. each node has been selected a logarithmic num- constraint (otherwise, the according dual nodes had to be neighbors in G). The analogous fact holds for primal ber of times. variables since dual nodes belonging to different local 3 We use p = 1/n1/k in the algorithm of Section 4 of [12], the LPs have distance at least 6 on G and thus primal properties then directly follow from Lemma 4.1 of [12]. nodes belonging to different local LPs have distance

at least 4 on G. Therefore, the local LPs do not interfere and together they form the sub-LPs induced by S (cf. Lemma 5.1). The complete LP approximation algorithm now consists of N independent parallel executions of the described basic algorithm. The variables of the N sub-LPs are added up and in the end, primal/dual nodes divide their variables by the maximum/minimum possible value to keep/make all constraints they occur in feasible.4 Finally, N can be chosen to optimize the approximation ratio.

6.1 The Cluster Graph The nodes v ∈ V in Gk are grouped into disjoint sets which are linked to each other as bipartite graphs. The structural properties of Gk are described using a directed cluster graph CGk = (C, A) with doubly labeled arcs ` : A → N × N. A node C ∈ C represents a cluster, i.e., one of the disjoint sets of nodes in Gk . An arc a = (C, D) ∈ A with `(a) = (δ c , δ d ) denotes that the clusters C and D are linked as a bipartite graph in which each node u ∈ C has degree δ c and each node v ∈ D has degree δ d . It follows that |C| · δ c = |D| · δ d . The cluster graph consists of two equal subgraphs, Theorem 5.1. Let N = αen1/k ln n for α ≈ 4.51. so-called cluster-trees CT as defined in [9]. In CG , we k k Executing the basic algorithm N times, summing up additionally add an arc `(C , C 0 ) := (1, 1) between two i i the variables of the N execution and dividing these corresponding nodes of the two cluster trees. Formally, sums as described, yields an αen1/k approximation of CT and CG are defined as follows. We call clusters k k (PP)/(DP) w.h.p. The algorithm requires O(k) rounds. adjacent to exactly one other cluster leaf-clusters, and Corollary 5.1. Using the network decomposition al- all other clusters inner-clusters. gorithm of [12], in only O(k) rounds, PP and DP Definition 6.1. [9] For a given δ and a positive intecan be approximated by a factor O(n1/k ) w.h.p. For ger k, the cluster tree CTk is recursively defined as k ∈ Θ(log n), this gives a constant factor approximafollows: tion in O(log n) rounds. CT1 := (C1 , A1 ), C1 := {C0 , C1 , C2 , C3 } 6 Lower Bound A1 := {(C0 , C1 ), (C0 , C2 ), (C1 , C3 )} We derive time lower bounds for distributed approxima`(C0 , C1 ) := (δ, δ 2 ), `(C0 , C2 ) := (δ 2 , δ 3 ), bility of packing problems, even in the LOCAL model. `(C1 , C3 ) := (δ, δ 2 ) More precisely, we prove lower bounds for the most basic packing problems, the fractional maximum match- Given CTk−1 , CTk is obtained in two steps: For ing problem (FMM). Our general approach follows [9] each inner-cluster Ci , add a new leaf-cluster C 0 with i in which similar results are obtained for minimum ver- `(Ci , C 0 ) := (δ k+1 , δ k+2 ). For each leaf-cluster Ci with i tex cover which is a covering problem. Specifically, our (Cp , Ci ) ∈ A and `(Cp , Ci ) = (δ p , δ p+1 ), add new packing lower bound graph is structurally similar (al- leaf-clusters C 0 with `(Ci , C 0 ) := (δ j , δ j+1 ) for j = j j though with subtle differences) to the one used in [9]. 1 . . . k + 1, j 6= p + 1. Let Ei denote the set of edges incident to node vi . FMM isP the natural LP relaxation Tk0 be two instances of P of MM and defined Definition 6.2. Let Tk and 0 as max ej ∈E yj , subject to CTk . Further, let Ci and Ci be corresponding clusters in vj ∈Ei yj ≤ 1, ∀vi ∈ V and yj ≥ 0, ∀ej ∈ E. The outcome of an edge’s decision Tk and Tk0 , respectively. We obtain the cluster graph (yj ) in a k-local computation is entirely based on the CGk by adding an arc `(Ci , Ci0 ) := (1, 1) for all clusters information gathered within its k-neighborhood. The Ci ∈ CTk . Further, we define n0 := |C0 ∪ C00 |. This idea for the lower bound is to construct a graph family uniquely defines the size of all clusters. Gk = (V, E) in which, after k rounds of communication, Figure 1 shows CT2 and CG2 . The shaded subtwo adjacent edges see exactly the same graph topology. graphs correspond to CT1 and CG1 , respectively, the Informally speaking, both of them are equally qualified dashed lines represent the links `(Ci , Ci0 ) := (1, 1). Note to join the matching. However, in Gk , taking the wrong decision will be ruinous and yields a suboptimal global that neither CTk nor CGk define the adjacency on the approximation. The construction of Gk is a two step level of nodes. They merely prescribe for each node process. First, the general structure of Gk is defined the number of neighbors in each cluster. We define 0 0 using the concept of a cluster-graph CGk . Secondly, S0 := C0 ∪ C0 and S1 := C1 ∪ C1 . The layer of a we construct an instance of Gk obeying the properties cluster is the distance to C0 in the cluster tree. Tk and Tk0 denote the two cluster trees constituting CGk . imposed by CGk . 4 The primal and dual variables x and y are divided by i j P 1 P minj∈Ni b1 ` aj` x` and maxi∈Nj c ` a`i y` , respectively. j

i

6.2 The Lower Bound Graph Gk Having defined the cluster graph CGk , it is now our goal to obtain

δ4

δ3

δ

δ2

C3 δ2

δ3

δ

δ2

δ4

δ3

C2 δ3

δ4

δ2

δ

δ2

C1 δ3

δ

δ2

Lemma 6.4. When applied to Gk = (V, E) as constructed in Subsection 6.2, any distributed, possibly randomized algorithm which runs for at most k rounds computes, in expectation, a solution of at most ALG ≤ |S0 |/(2δ 2 ) + (|V | − |S0 |).

C0

C0

6.3 Analysis We now derive the lower bounds on the approximation ratio for k-local FMM algorithms. Let OPT be the optimal solution for FMM and let ALG be the solution computed by any algorithm. All nodes in S0 and S1 have the same view and therefore, every edge in E 0 sees the same topology Ve,k .

C’0

Figure 1: Cluster-Tree CT2 and Cluster-Graph CG2 .

a realization of Gk which has the structure imposed by CGk and features the additional property that there are no short cycles. As we must prove that the topologies seen by nodes in S0 and S1 are identical, the absence of short cycles is of great help. Particularly, if there are no cycles of length 2k + 1 and less, all nodes see a tree locally. The girth of a graph G, denoted by g(G), is the length of the shortest cycle in G. Lemma 6.1 states that it is indeed possible to construct Gk as described above. Lemma 6.1. If k + 1 ≤ δ/2, Gk can be constructed such that the following conditions hold: (I) Gk follows the structure of CGk . (II) The girth of Gk is at least g(Gk ) ≥ 2k + 1. 2 (III) Gk has n ≤ 42k δ 4k nodes. Next we show that all nodes in S0 and S1 have the same view and consequently, all edges in E 0 see the same topology. Using the following result from [9] facilitates this task. Lemma 6.2. [9] Let Gk be an instance of a cluster tree CTk with girth g(Gk ) ≥ 2k + 1. The views of all nodes in clusters C0 and C1 are identical up to distance k.

Proof. The fractional value assigned to ei = (u, v) by an algorithm is denoted by yi . The decision of which value yi is assigned to edge ei depends only on the view the topologies Tu,k and Tv,k and the labelings L(Tu,k ) and L(Tv,k ), which ei can collect during the k communication rounds. Assuming that the labeling of Gk is chosen uniformly at random, the labeling L(Tu,k ) for any node u is also chosen uniformly at random. All edges connecting nodes in S0 and S1 see the same topology. If the labels are chosen uniformly at random, it follows that the distribution of the views and therefore the distribution of the yi is the same for all those edges. We call the random variables describing the distribution of the yi , Yi . Let u ∈ S1 be a node of S1 . The node u has δ 2 neighbors in S0 . Therefore, for edges ei between nodes in S0 and S1 , by linearity of expectation, E [Yi ] ≤ 1/δ 2 because otherwise there exist labelings for which the calculated solution is not feasible. By Lemma 6.3, edges ej with both end-points in S0 have the same view as edges between S0 and S1 . Hence, also for the value yj of ej , E [Yj ] ≤ 1/δ 2 must hold. There are |S0 |/2 such edges and therefore the expected total value contributed by edges between two nodes in S0 is at most |S0 |/(2δ 2 ). All edges which do not connect two nodes in S0 , have one end-point in V \ S0 . In order to get a feasible solution, the total value of all edges adjacent to a set of nodes V 0 , can be at most |V 0 |. This can for example be seen by looking at the dual problem, a kind of minimum vertex cover where some edges only have one end node. Taking all nodes of V 0 (assigning 1 to the respective variables) yields a feasible solution for this vertex cover problem. The claim now follows by applying Yao’s minimax principle.

Because Gk has girth at least 2k + 1 by Lemma 6.1, the two cluster-trees Tk and Tk0 constituting Gk must have girth 2k + 1 as well. It follows from Lemma 6.2 that the desired equality of views holds for both Tk and Tk0 . We now derive the lower bound. Lemma 6.4 gives an Based on this fact, it is now easy to show that equality upper bound on the number of nodes chosen by any klocal FMM algorithm. Choosing all edges within S0 is of views holds in Gk , too. feasible, hence, |OPT | ≥ |S0 |/2. In order to establish Lemma 6.3. Let Gk be an instance of a cluster graph a relationship between n, |S0 |, δ, and k, we bound n k+1 ) using a geometric series. The CGk with girth g(Gk ) ≥ 2k + 1. The views of all nodes as n ≤ |S0 |(1 + δ−(k+1) in clusters S0 and S1 are identical up to distance k. second lower bound then follows easily from ∆ = δ k+2 .

Theorem 6.1. For all pairs (n, k) and (∆, k), there are graphs G and a constant c ≥ 1/4, such that in k communication rounds, every distributed algorithm for 2 FMM ¡on G has¢ approximation ratios at least Ω(nc/k /k) 1/k and Ω ∆ /k , respectively. p By setting k = β log n/ log log n and k = β log ∆/ log log ∆, respectively, for a constant β > 0, we obtain the following corollary. Corollary 6.1. In order to obtain a polylogarithmic or constant approximation ratio, every p distributed algorithm for FMM requires at least Ω( log n/ log log n) and Ω(log ∆/ log log ∆) communication rounds. The same lower bounds hold for the construction of maximal matchings and maximal independent sets. Remark: The algorithm in Section 5 achieves a polylogarithmic approximation in O(log ∆/ log log ∆) communication rounds. Therefore, for polylogarithmic approximations, our lower bound for FMM is tight. 7 Conclusions It is interesting to view local computation in a wider context of computational models. Approximation algorithms and online algorithms try to bound the degradation of a globally optimal solution caused by limited computational resources and knowledge about the future, respectively. More recently, the “price of anarchy,” has been proposed to measure the suboptimality resulting from selfish individuals [20]. In a similar spirit, our paper sheds light on the price of locality, i.e., the degradation of a globally optimal solution if each individual’s knowledge is restricted to its neighborhood or local environment. Specifically, the upper and lower bounds presented in this paper characterize the achievable trade-off between local information and the quality of a global solution of covering and packing problems. References [1] N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567–583, 1986. [2] Y. Bartal, J. W. Byers, and D. Raz. Global Optimization Using Local Information with Applications to Flow Control. In Proc. of the 38th Symp. on Foundations of Computer Science (FOCS), pages 303–312, 1997. [3] D. Dubhashi, A. Mei, A. Panconesi, J. Radhakrishnan, and A. Srinivasan. Fast Distributed Algorithms for (Weakly) Connected Dominating Sets and Linear-Size Skeletons. In Proc. of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 717–724, 2003. [4] M. Elkin. Unconditional Lower Bounds on the TimeApproximation Tradeoffs for the Distributed Minimum Spanning Tree Problem. In Proc. of the 36th ACM Symposium on Theory of Computing (STOC), 2004.

[5] F. Fich and E. Ruppert. Hundreds of impossibility results for distributed computing. Distributed Computing, 16(2-3):121–163, 2003. [6] L. Fleischer. Approximating Fractional Multicommodity Flow Independent of the Number of Commodities. SIAM Journal on Discrete Math., 13(4):505–520, 2000. [7] A. Israeli and A. Itai. A Fast and Simple Randomized Parallel Algorithm for Maximal Matching. Information Processing Letters, 22:77–80, 1986. [8] L. Jia, R. Rajaraman, and R. Suel. An Efficient Distributed Algorithm for Constructing Small Dominating Sets. In Proc. of the 20th Symposium on Principles of Distributed Computing (PODC), pages 33–42, 2001. [9] F. Kuhn, T. Moscibroda, and R. Wattenhofer. What Cannot Be Computed Locally! In Proc. of the 23rd ACM Symp. on Principles of Distributed Computing (PODC), pages 300–309, 2004. [10] F. Kuhn and R. Wattenhofer. Constant-Time Distributed Dominating Set Approximation. In Proc. of the 22nd ACM Symposium on Principles of Distributed Computing (PODC), pages 25–32, 2003. [11] N. Linial. Locality in Distributed Graph Algorithms. SIAM Journal on Computing, 21(1):193–201, 1992. [12] N. Linial and M. Saks. Low Diameter Graph Decompositions. Combinatorica, 13(4):441–454, 1993. [13] M. Luby. A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 15:1036–1053, 1986. [14] M. Luby and N. Nisan. A Parallel Approximation Algorithm for Positive Linear Programming. In Proc. of the 25th ACM Symposium on Theory of Computing (STOC), pages 448–457, 1993. [15] M. Naor and L. Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259– 1277, 1995. [16] C. Papadimitriou and M. Yannakakis. Linear Programming without the Matrix. In Proc. of the 25th Symp. on Theory of Computing (STOC), pages 121–129, 1993. [17] D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. [18] S. Plotkin, D. Shmoys, and E. Tardos. Fast Approximation Algorithms for Fractional Packing and Covering Problems. Mathematics of Operations Research, 20:257–301, 1995. [19] S. Rajagopalan and V. Vazirani. Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs. SIAM Journal on Computing, 28:525–540, 1998. [20] T. Roughgarden and E. Tardos. How Bad is Selfish Routing? In Proc. of the 41 th Symp. on Foundations of Computer Science (FOCS), pages 93–102, 2000. [21] N. Young. Randomized Rounding without Solving the Linear Program. In Proc. of the 6th Symposium on Discrete Algorithms (SODA), pages 170–178, 1995. [22] N. Young. Sequential and Parallel Algorithms for Mixed Packing and Covering. In Proc. of the 42 nd Symposium on Foundations of Computer Science (FOCS), pages 538–546, 2001.

Abstract Achieving a global goal based on local information is challenging, especially in complex and large-scale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible tradeoff between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρ∆)1/k -approximation for general covering and packing problems in time O(k2 ), where ρ depends on the LP’s coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n1/k ) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its k-neighborhood.

1 Introduction Many of the most fascinating and fundamental systems in the world are large and complex networks, such as the human society, the Internet, or the human brain. Such systems have in common that their entirety is composed of a multiplicity of individual entities; human beings in society, hosts in the Internet, or neurons in the brain. As diverse as these systems may be, they share the key characteristic that the capability of direct communication of each individual entity is restricted to only a small subset of neighboring entities. Most human communication, for instance, is between acquaintances or within the family, and neurons are directly linked with merely a relatively small number of other neurons for neurotransmission. On the other hand, in spite of each node thus being inherently “near-sighted,” i.e., restricted to local communication, the entirety of the system is supposed to come up with some kind of global solution, or to keep an equilibrium. Achieving a global goal based on local information is challenging. Many of the systems which are the focus of computer science fall exactly into the above mentioned category of networks. In the Internet, largescale peer-to-peer systems, or mobile ad hoc and sensor networks, no node in the network is capable of keeping

global information on the network. Instead, these nodes have to perform their intended (global) task based on local information only. In other words, all computation in these systems is inherently local! Not surprisingly, studying the fundamental possibilities and limitations of local computation is therefore of interest to theoreticians in approximation theory, distributed computing, and graph theory. The study of local computation has been initiated by the pioneering work of Linial [11], and Naor and Stockmeyer [15] more than a decade ago. Also, the work of Peleg [17] has resulted in numerous interesting and deep results. But nonetheless, there remains a great number of important open problems related to questions such as what kind of global tasks can be performed by individual entities that have to base their decisions on local information, or how much local information is required in order to come up with a globally optimal solution. For instance, the open question from [16], that is, characterizing the trade-off between communication among agents to exchange information and the global utility achieved, has been unanswered. It is the goal of this paper to make a step towards answering this open problem and, more generally, to bring forward some of the underlying principles and trade-offs governing local computation. Not surprisingly, many global criteria such as counting the total number of nodes in the network or obtaining a minimum spanning tree cannot be met if every node’s decision is based solely on local knowledge. On the other hand, many fundamental coordination tasks and applications in large-scale networks appear to be easier to handle from a “local-global” perspective. Specifically, classic graph theory problems such as dominating set or matching can be formulated as standard covering and packing problems. The nature of simple covering and packing problems like minimum vertex cover or maximum matching appears to be local and intuitively, one may think that each node’s (edge’s) decision is not affected by very distant nodes (edges). Interestingly, we prove in this paper that this intuition is misleading even for the most basic packing problems.

On the positive side, we show that there exist distributed approximation algorithms that almost achieve the optimal trade-off, even in the practically important case in which the amount of information exchanged in each message is limited. Specifically, we give the following results: • Consider a network with n nodes and maximum degree ∆. Assume that each node in a network graph has to base its decision on its k-hop neighborhood. We present an efficient deterministic distributed algorithm that operates with small messages of size O(log n) bits. The algorithm achieves a (ρ∆)1/k -approximation for general covering and packing problems in O(k 2 ) communication rounds, where ρ depends on the coefficients of the underlying LP. • When message size is unbounded, each network node can easily gather the entire information from its O(k)-neighborhood in O(k) communication rounds. Hence, in this case, the achievable trade-off is a true consequence of locality restriction only. We present an algorithm producing an O(n1/k )-approximation if each node knows its O(k)-neighborhood. • In combination with (distributed) randomized rounding, the above algorithms can be transformed into constant-time distributed algorithms having non-trivial approximation ratios for various combinatorial problems.

message model, respectively. The packing lower bound is derived in subsequent Section 6. Section 7 concludes the paper. Due to lack of space, some proofs are omitted from this extended abstract.1 2 Related Work Little is known about the fundamental limitations of locality-based approaches. Fich and Ruppert, for instance, describe a numerous lower bounds and impossibility results in distributed computing [5]. But most of them apply to other computational models where locality is no issue or there are additional, more restrictive limiting factors, such as bounded message size [4]. There have been virtually no nontrivial lower bounds for local computation, besides Linial’s seminal Ω(log∗ n) time lower bound for constructing a maximal independent set on a ring [11]. In addition, we have shown that minimum vertex cover and thus covering prob2 lems cannot be approximated better than Ω(nc/k /k) and Ω(∆1/k/k) if each node’s information is restricted to its k-neighborhood [9]. On the positive side, it was shown by Naor and Stockmeyer [15] that there exist locally checkable labelings which can be computed in distributed constant time, i.e., with completely local information only. The focus of this paper is to understand locality in problems that can be formulated as packing and covering LPs. There are a number of (parallel) algorithms for solving such LPs which are faster than interior-point methods that can be applied to general LPs (e.g. [6, 14, 18, 22]). All these algorithms need at least some global information to work. The problem of approximating positive LPs using only local information has been introduced in [16]. The first algorithm achieving a constant approximation in polylogarithmic time is described in [2]. Distributed algorithms targeted for specific covering and packing problems include algorithms for the minimum dominating set problem [3, 8, 19] as well as algorithms for maximal matchings and maximal independent sets [1, 7, 13]. This implies a constant approximation for maximum matching. All described distributed algorithms have a time complexity which is at least logarithmic in n. That is, each node may gather information which is as far away as O(log n) hops. Hence, while these algorithms provide solutions to particular problems, they do not fully explore the trade-off between local knowledge and solution quality. A distributed algorithm for minimum dominating set running in an arbitrary, possibly constant number of rounds is found in [10].

• Finally, we show that the trade-off achieved by our algorithms is almost tight. Specifically, we prove that even the most simple packing problem, (fractional) maximum matching, cannot be ap2 1/k proximated within Ω(nc/k /k) p and Ω(∆ /k), respectively. This implies Ω( log n/ log log n) and Ω(log ∆/ log log ∆) time lower bounds for (possibly randomized) distributed algorithms in order to obtain a constant or polylogarithmic approximation for maximum matching and packing problems, even if message size is unbounded. This lower bound extends a similar result that has recently been achieved for the minimum vertex cover problem in [9]. Note that by giving upper and lower bounds for general covering and packing LPs, we show that many different natural problems behave similarly with regard to local approximability. This is of great theoretical interest since such a classification of problems may provide a completely new insight into the impact of locality on algorithms. Related work and the model of computation are described in Sections 2 and 3. In subsequent Sections 4 1 A version containing all proofs can be found as TIK techniand 5, we give distributed algorithm for general covercal report 229 at ftp://ftp.tik.ee.ethz.ch/pub/publications/TIKing and packing LPs in the bounded and unbounded Report229.pdf

3 Model We describe the network as an undirected graph G = (V, E). The vertices V = {v1 , . . . , vn } represent the network entities or nodes (e.g. processors) and the edges represent bidirectional communication channels. We distinguish two prototypical and classic message passing models [17], LOCAL and CON GEST , depending on how much information can be sent in each message. In the LOCAL model (e.g., [11, 15, 17]), knowing your k-neighborhood and performing k communication rounds are equivalent. It is assumed that in every communication round, each node in the network can send an arbitrarily long message to each of its neighbors. Local computations are for free. Each node has a unique identifier and initially, nodes have no knowledge about the network graph. In k communication rounds, a node v may collect the IDs and interconnections of all nodes up to distance k from v, because messages are unbounded. Hence, each node has a partial (local) view of the graph; it knows its entire vicinity up to distance k. Let Tv,k be the topology seen by v after k rounds. Tv,k is the graph induced by the k-neighborhood of v without all edges between nodes at distance exactly k. The labeling (i.e., the assignment of IDs) of Tv,k is denoted by L(Tv,k ). The view of a node v is the pair, Vv,k := (Tv,k , L(Tv,k )). The view of an edge e = (u, v) is the union of views of its incident nodes. The best a local algorithm can do in time k, is to have every node v collect its k-neighborhood and base its decision on Vv,k . Since the LOCAL model abstracts away other aspects arising in the design of distributed algorithms (congestion, fast local computation, . . . ), it is the most fundamental model when studying the phenomenon of locality; particularly for lower bounds. In practice, the amount of information exchanged between two neighbors in one communication step is limited. The CON GEST model [4, 17] takes into account the volume of communication. This model limits the information that can be sent in one message to O(log n) bits. Given this additional restriction, even problems on the complete network graph, which could be solved optimally in a single communication round in the LOCAL model, become nontrivial. A fractional covering problem (PP) and its dual fractional packing problem (DP), are linear programs of the canonical forms min s.t.

cT x A·x≥b x≥0

max s.t.

bT y AT · y ≤ c y≥0

where all aij , bi , and ci are non-negative. We will use

the term primal LP (PP) for the minimization and dual LP (DP) for the maximization problem. The number of primal and dual variables are denoted by m and n, respectively. Let amax := maxi,j {aij , bi , ci } be the maximum coefficient and amin := mini,j {aij , bi , ci } \ {0} be the minimum non-zero coefficient of (PP) and (DP). ρ := amax /amin is the maximum ratio between any two coefficients. Analogously to [2, 16], we consider the following distributed setting. The linear program is bound to a network graph G = (V, E). Each primal variable xi and (p) each dual variable yj is associated with a node vi ∈ V (d) and vj ∈ V , respectively. There are communication links between primal and dual nodes wherever the respective variables occur in the corresponding inequality. (p) (d) Thus, (vi , vj ) ∈ E if and only if xi occurs in the j th (p)

inequality of (PP), i.e., vi 2

aji > 0 . The degrees of (d) δj ,

(d)

and vj

are connected iff

(p) vi

(d) (p) and vj are called δi (p) maxi δi and ∆d := maxj

and (d)

respectively. ∆p := δj are called the primal and dual degree, respectively. The (p) (p) set of dual neighbors of vi is denoted by Ni , the set (d) (d) of primal neighbors of vi by Ni . Where convenient, (p) (d) Ni and Nj also denote the sets of the indices of the respective nodes. 4 Bounded Messages In this section, we describe an efficient distributed algorithm to approximate covering and packing linear programs in the CON GEST model. For our algorithm, we need the LPs (PP) and (DP) to be of the following special form: (4.1)

∀i, j : bi = 1,

aij = 0 or aij ≥ 1.

The transformation to (4.1) is done in two steps. First, every aij is replaced by a ˆij := aij /bi and bi is replaced by 1. In the second step, the ci and a ˆij are divided by λi := minj {ˆ aji } \ {0}. The optimal objective values of the transformed LPs are the same. A feasible solution for the transformed LP (4.1) can be converted to a feasible solution of the original LP by dividing all x-values by the corresponding λi and by dividing the y-values by the corresponding bi . This conserves the values of the objective functions. Note that the described transformation can be computed locally in a constant number of rounds. For the rest of this section, 2 Note that in order to solve such a problem in a real network setting where only some variables correspond to nodes, the other variables may be simulated by the nodes as well. Variables associated to edges (like in vertex cover or maximum matching) can be simulated by incident nodes.

LP Approximation (p) Algorithm for Primal Node vi :

LP Approximation (d) Algorithm for Dual Node vi : yi := yi+ := wi := fi := 0; ri := 1; for ep := kp − 2 to −f − 1 by −1 do for 1 to h do r˜i := ri ; 5: for ed := kd − 1 to 0 by −1 do

xi := 0; for ep := kp − 2 to −f − 1 by −1 do for 1 to h do P (∗ γi := cmax j aji rj ∗) ci 5: for ed := kd − 1 to 0 by −1 do P a r ˜ 6: γ˜i := cmax j ji j ; ci 1: 2: 3: 4:

7: 8: 9: 10:

if

e /k γ˜i ≥ Γpp p then ed /kd x+ ; xi i := 1/Γd

1: 2: 3: 4: 6: 7:

x+ i ;

:= xi + fi; send x+ i , γ˜i to dual neighbors;

8: 9: 10:

11:

11:

12:

12:

13: 14: 15: 16: 17:

13: 14: 15: 16: 17:

receive r˜j from dual neighbors od;

receive rj from dual neighbors od od; P 21: xi := xi / minj∈N (p) ` aj` x` 18: 19: 20:

18: 19: 20: 21:

i

receive x+ j , γ˜jPfrom primal neighbors; yi+ := yi+ + r˜i j aij x+ j /γ˜j ; P wi+ := j aij x+ ; j wi := wi + wi+ ; fi := fi + wi+ ; if wi ≥ 1 then r˜i := 0 fi; send r˜i to primal neighbors od; increase duals(); send ri to primal neighbors od od; P yi := yi / maxj∈N (d) c1j ` a`j y` i

Algorithm 1: Distributed LP Approximation Algorithm

we assume that the coefficients of the LP are given according to (4.1). We start the description of the algorithm with a general outline. As our algorithm borrows from the greedy dominating set/set cover algorithm, it is useful to view the distributed LP algorithm in this context. The greedy minimum dominating set (MDS) algorithm starts with an empty set and sequentially adds the node which covers the most not yet covered nodes. The LP relaxation of MDS asks for variables xi for the nodes vi such that the sum of the xi in the 1-neighborhood of every node is at least 1. Analogous to the sequential greedy approach, we also start with all xi set to 0 and we give priority to nodes with many uncovered neighbors when increasing the xi . In particular, we always increase the xi of all the nodes whose number of uncovered neighbors is maximum up to a certain factor (active nodes). In order not to ‘over-cover’ a node with many active neighbors, we have to carefully choose the increment of the xi at active nodes. As we proceed, we simultaneously compute a solution for the dual LP such that the objective values of the solutions stay the same. In the end, each node is covered at least f times and each dual constraint is fulfilled up to a factor αf . Hence by dividing by f and αf , we obtain feasible, α-

approximate primal and dual solutions, respectively. In order to achieve that every primal inequality is (d) fulfilled f times, each dual node vi needs a requirement ri ≤ 1 which is decreased every time the corresponding primal constraint is achieved and a variable fi which counts how many times the primal constraint has been fulfilled (cf. [21]). The decision whether a primal node (p) vi is active and can increase xi is based on the efficiency per cost ratio γi which is defined as follows: γi :=

cmax X aji rj . ci j

For simplicity, we assume that all nodes know cmax := max{ci } as well as two other global quantities Γp and Γd which are defined as Γp := max i

n cmax X · aji ci j=1

and Γd := max i

m X

aij .

j=1

At the price of a considerably more complicated (and less readable) algorithm, it is possible to get rid of this assumption. For details, we refer to the full paper. The detailed algorithm is given by Algorithm 1 along with the procedure increase duals() which is

(p)

Lemma 4.3. Let vi be a primal node and let Yi := P a y j ji j be the weighted sum of the y-values of its dual neighbors. Further, let Yi+ be the increase of Yi and γi− be the decrease of γi during an execution of increase duals(). We have

procedure increase duals(): 1: if wi ≥ 1 then 2: if fi ≥ f then 3: yi := yi + yi+ ; yi+ := 0; 4: ri := 0; wi := 0 5: else if wi ≥ 2 then 6: yi := yi + yi+ ; yi+ := 0; bw c/k 7: ri := ri /Γp i p 8: else 1/k 1/k 9: λ := max{Γd d , Γp p }; e /k 10: yi := yi + min{yi+ , ri λ/Γpp p }; e /k 11: yi+ := yi+ − min{yi+ , ri λ/Γpp p }; 1/k 12: ri := ri /Γp p 13: fi; 14: wi := wi − bwi c 15: fi

3/kp

Yi+ ≤

Γp

1/kp

· max{Γp 1/k γi (Γp p

1/kd

, Γd

}

− 1)

·

ci cmax

· γi− .

Proof. We prove the lemma by showing that the in(p) (d) equality holds for every dual neighbor vj of vi . Let βj be the increase of yj and let rj− be the decrease of rj . We show that 1/kp

(4.3)

βj ≤

1/kp

Γp

1/kd

· max{Γp

, Γd

e /k 1/k Γpp p (Γp p

− 1)

}

· rj− . (e +2)/kp

used by the dual nodes. The algorithm has two parameters kp ≥ 1 and kd ≥ 1 which determine the tradeoff between time complexity and approximation quality. The bigger kp and kd , the better the approximation ratio of the algorithm. On the other hand, smaller kp and kd lead to a faster algorithm. Algorithm 1 also makes use of two values f and h which are defined as follows: & ' & ' kp + 1 kp f := and h := 1 + 1/k . 1/k Γp p − 1 Γp p ln Γp In the following, we present lemmas which establish all the necessary details to analyze Algorithm 1. The goal of the outer ep -loop is to reduce the maximum “weighted primal degree” γi . This is reflected by the following lemma. (p)

The lemma then follows because γi ≤ Γp p (Lemma 4.1) and because X cmax X Yi+ = aji βj and γi− = aji rj− . c i j j

To prove Inequality (4.3), we again consider the cases where wj ≥ 2 and where 1 ≤ wj < 2. If wj ≥ 2, e /k 1/k by Lemma 4.2, βj = yj+ ≤ rj (1 + Γd d )/Γpp p . The 2/k

requirement rj is divided by at least Γp p and therefore 2/k 2/k rj− ≥ rj (Γp p − 1)/Γp p . Together, we get 2/kp

1/kd

βj

≤

1 + Γd ³

≤

·

e /k Γpp p

1+

1/k Γp p

e /kp

2/k Γp p

´

³

Γpp

Γp

−1

· rj−

1/k Γp p 1/kp

Γp

1/k

1/k

max{Γp p , Γd d } ´³ ´ rj− . 1/kp + 1 Γp −1

Lemma 4.1. For each primal node vi , at all times (e +2)/kp during Algorithm 1, γi ≤ Γp p .

For 1 ≤ wj < 2, the proof is along the same lines. Here, 1/k e /k 1/k 1/k βj ≤ rj max{Γp p , Γd d }/Γpp p and rj− = rj (Γp p −

One complete run (kd iterations) of the innermost ed -loop can be seen as one parallel greedy step. Primal nodes with large γi increase their xi such that the corresponding increases yi+ of the dual variables are almost feasible.

1)/Γp

Lemma 4.2. Each time a dual node enters increase duals() in Algorithm 1, (4.2)

yi+ ≤ ri ·

wi e /kp

Γpp

and yi+ ≤ ri ·

1/kd

Γd

+1

e /kp

Γpp

1/kp

. Again, we obtain Inequality (4.3): 1/kp

βj ≤

max{Γp

e /kp

Γpp

1/kd

, Γd

}

1/kp

·

Γp

1/kp

Γp

−1

· rj− .

We do not have to consider the case fj ≥ f explicitly because the same analysis as for wj ≥ 2 applies in this case. (p)

.

As shown in Lemma 4.4, all the increases of the dual variables together render the dual constraints feasible up to a small factor times (kp + f + 1). We first need the following helper lemma.

Lemma 4.4. Let vi be a primal node and Yi = P j aji yj be the weighted sum of the y-values of the dual (p)

neighbors of vi . After the main part of the algorithm (i.e., after the loops at line 20), n o ci 1/kd 1/kp p (kp + f + 1)Γ3/k max Γ , Γ . Yi ≤ p p d cmax

Proof. For simplicity, we define Q :=

1 3/kp 1/k p Γ max{Γ1/k , Γd d }. p cmax p

Before γi is decreased for the last time, we have γi ≥ (f −1)/kp 1/Γp because at least one rj in the dual neigh(p) borhood of vi has to be greater than 0. If we assume that the last time γi is decreased it is only reduced to (f +1)/kp γi = 1/Γp , Lemma 4.3 still holds. The analysis is exactly the same as for the case wj ≥ 2 in Lemma 4.3. By Lemma 4.3, Yi is therefore bounded by the 1/k area under the curve ci Q/(Γp p − 1) · 1/x for x between (f +1)/kp 1/Γp and Γp : Z Γp ci Q 1 · dx Yi ≤ 1/k p 1 x Γp −1 (f +1)/k

Because fj ≥ f , all yj+ are 0 in the end and thus yj is equal to the sum of all increases of yj+ . ¤ Combining the above lemmas, we get the following theorem. Theorem 4.1. For arbitrary kp , kd ≥ 1, Algorithm 1 approximates (PP) and (DP) by a factor n o 1/k p p Γ4/k max Γ1/k , Γd d . p p The time complexity of Algorithm 1 is Ã Ã !Ã !! 1 kp O kd kp 1 + 1/k 1 + 1/k . Γp p − 1 Γp p log Γp For kp ∈ O(log Γp ), this simplifies to O(kd kp ).

p

Proof. For the approximation ratio, we have to look at line 21 of Algorithm 1 where all x and y values = ≤ ci (kp + f + 1)Q. are divided by the largest possible values to keep/make 1/k Γp p − 1 the primal/dual solution feasible. By Lemma 4.5, each primal constraint is satisfied at least f times. Therefore, The last inequality follows from ln(1 + t) ≤ t. ¤ all primal variables are divided by at least f . Due to At the end of the algorithm, all primal constraints Lemma 4.4, for each primal node, the sum of the y are satisfied at least f times. Further, the primal and values of its dual neighbors is at most ci (kp +f +1)Q for Q as defined in Lemma 4.4. Dividing all dual variables dual objective functions are the same. by (kp + f + 1)Q therefore renders the dual solution Lemma 4.5.PAfter the loops P at line 20, ∀i: ri = 0 and feasible. By Lemma 4.5, the ratio between the objective m n functions of the primal and the dual solutions is fi ≥ f and i=1 ci xi = cmax j=1 yj . Pm kp + f + 1 Proof. When entering the ep -loop for the last time, by Pi=1 ci xi ≤ c Q max n f y Lemma 4.1, j=1 i X X kp +1 +1 kp + 1/k Γp(−f +1)/kp ≥ γj ≥ aij ri ≥ ri . Γp p −1 ≤ c Q max (p) i kp +1 Γp

³ ´ 1/k ci (kp + f + 1)Q ln Γp p

i∈Nj

γj can only be greater than 0 if there is exactly one ri (p) in the dual neighborhood of vj which is greater than zero. If ri is still greater than 0 when ed = 0, xj will be increased by 1 which makes wj ≥ 1 and therefore ri = 0 after the next call to increase duals(). fi counts the number of times the ith constraint of (PP) is satisfied. It is increased together with wi in line 13 of Algorithm 1. Every time the integer part of wi bw c/k is increased, ri is divided by Γp i p and wi is set to wi − bwi c. Therefore, ri = 0 implies fi ≥ f . (p) Let vi be a primal node which increases xi by x+ i (d) (p) (line 8). All dual neighbors vj of vi increase yj+ by + aji r˜j x+ i /γ˜i . Hence, the sum of the yj -increases over all (p)

dual neighbors of vi

is

P X ci + x+ j aji r˜j + i = x . aji r˜j = xi cmax P γ˜i j cmax i j aji r˜j ci

1/kp

Γp

=

p Q cmax Γ1/k p

−1

o n 1/kd 1/kp p max Γ , Γ . = Γ4/k p p d

Because of the duality theorem for linear programming, this ratio is an upper bound on the approximation ratio for (PP) and (DP). As for the time complexity, note that each iteration of the inner-most loop (ed -loop) requires two rounds. Hence, the algorithm has time complexity O(kd (kp + f )h). The claim follows from substituting the actual values for f and h. For kp ∈ O(log Γp ), 1/k Γp p −1 is a constant and therefore the time complexity simplifies to O(kd kp ). ¤ Corollary 4.1. For sufficiently small ε, Algorithm 1 computes a (1 + ε)-approximation for (PP) and (DP) ¡ ¢ in O log Γp log Γd /ε4 rounds. In particular, a constant factor approximation can be achieved in time O(log Γp log Γd ).

Remark: Using methods similar to the ones described in [2, 14], it is possible to get rid of the dependency on the coefficients ρ := amax /amin . As a result, the running time and approximation ratio would depend on the number of nodes m and n instead of the degrees ∆p and ∆d . Distributed Randomized Rounding We can apply our distributed LP approximation algorithms together with standard distributed randomized rounding techniques to obtain distributed approximation algorithms for a number of combinatorial problems. We can prove that given an α-approximate solution for the LP relaxation of problems for which the matrix elements aij ∈ {0, 1}, we can compute in a constant number of rounds a O(α log ∆p )-approximation for the corresponding covering IP and a O(α∆d )-approximation for the packing IP.

For the decomposition of (PP) and (DP), we need the following lemma. 0 Lemma 5.1. Let {y10 , . . . , ym 0 } be a subset of the dual 0 variables of DP and let x1 , . . . , x0n0 be the primal variables which are adjacent to the given subset of the dual variables. Further let P P 0 and DP 0 be LPs where the matrix A0 consists only of the columns and rows corresponding to the variables in x0 and y 0 . Every feasible solution for P P 0 makes the corresponding primal inequalities in P P feasible and every feasible solution for DP 0 is feasible for DP (variables not occurring in P P 0 and DP 0 are set to 0). Further, the values of the objective functions for the optimal solutions of P P 0 and DP 0 are upper bounded by the optimal values for P P and DP .

We call P P 0 and DP 0 the sub-LPs induced by the subset 0 We apply the graph {y10 , . . . , ym 0 } of dual variables. decomposition algorithm of [12] to obtain P P 0 and DP 0 5 Unbounded Messages (as in Lemma 5.1) which can be solved locally. In [12], Linial and Saks presented a randomized disFor the decomposition of the linear program, we tributed algorithm to decompose a graph into sub- define G such that the node set V is the set of dual graphs of limited diameter. We use their algorithm to nodes of the graph G and the edge set E is decompose the linear program into sub-programs which ¯ © ª can be solved locally in the LOCAL model. For a genE := (u, v) ¯ u, v ∈ V ∧ dG (u, v) ≤ 4 . eral graph G = (V, E) with n nodes, the algorithm of [12] yields a subset S ⊆ V of V such that each node By this, we can guarantee that non-adjacent nodes in u ∈ S has a leader `(u) ∈ V and such that the following G do not have neighboring primal nodes in G whose variables occur in the same constraint of (PP). Further, properties hold.3 a message over an edge of G can be sent in 4 rounds (I) ∀u ∈ S : d(u, `(u)) < k on the network graph G. The basic algorithm for a (II) ∀u, v ∈ S : `(u) 6= `(v) −→ (u, v) 6∈ E. dual node v to approximate P P and DP then works as (III) S can be computed in k rounds. follows: (IV) ∀u ∈ V : Pr[u ∈ S] ≥ en11/k . 1: Run graph decomposition of [12] on G; 2: if v ∈ S then d(u, v) denotes the distance between two nodes u and 3: send IDs of primal neighbors to `(v). v on G. We apply the algorithm of [12] to obtain con4: fi; nected components of G with the following properties. 5: if v = `(u) for some u ∈ S then (I) The components have small diameter. 6: compute local PLP/DLP (cf. Lemma 5.1) (II) Different components are far enough from each of variables of u ∈ S for which v = `(u). other such that we can define a local linear program 7: send resulting values to nodes holding the for each component in a way in which the LPs of respective variables. any two components do not interfere. 8: fi (III) Each node belongs to one of the components with The primal nodes only forward messages in steps 1, probability at least p, where p depends on the 3, and 7 and receive the values for their variables in diameter we allow the components to have. step 7. We now have a closer look at the locally Because of the limited diameter, the LPs of each com- computed LPs in line 6. By Property (II) of the graph ponent can then be computed locally. We apply the de- decomposition algorithm, dual variables belonging to composition process in parallel often enough such that different local LPs cannot occur in the same dual w.h.p. each node has been selected a logarithmic num- constraint (otherwise, the according dual nodes had to be neighbors in G). The analogous fact holds for primal ber of times. variables since dual nodes belonging to different local 3 We use p = 1/n1/k in the algorithm of Section 4 of [12], the LPs have distance at least 6 on G and thus primal properties then directly follow from Lemma 4.1 of [12]. nodes belonging to different local LPs have distance

at least 4 on G. Therefore, the local LPs do not interfere and together they form the sub-LPs induced by S (cf. Lemma 5.1). The complete LP approximation algorithm now consists of N independent parallel executions of the described basic algorithm. The variables of the N sub-LPs are added up and in the end, primal/dual nodes divide their variables by the maximum/minimum possible value to keep/make all constraints they occur in feasible.4 Finally, N can be chosen to optimize the approximation ratio.

6.1 The Cluster Graph The nodes v ∈ V in Gk are grouped into disjoint sets which are linked to each other as bipartite graphs. The structural properties of Gk are described using a directed cluster graph CGk = (C, A) with doubly labeled arcs ` : A → N × N. A node C ∈ C represents a cluster, i.e., one of the disjoint sets of nodes in Gk . An arc a = (C, D) ∈ A with `(a) = (δ c , δ d ) denotes that the clusters C and D are linked as a bipartite graph in which each node u ∈ C has degree δ c and each node v ∈ D has degree δ d . It follows that |C| · δ c = |D| · δ d . The cluster graph consists of two equal subgraphs, Theorem 5.1. Let N = αen1/k ln n for α ≈ 4.51. so-called cluster-trees CT as defined in [9]. In CG , we k k Executing the basic algorithm N times, summing up additionally add an arc `(C , C 0 ) := (1, 1) between two i i the variables of the N execution and dividing these corresponding nodes of the two cluster trees. Formally, sums as described, yields an αen1/k approximation of CT and CG are defined as follows. We call clusters k k (PP)/(DP) w.h.p. The algorithm requires O(k) rounds. adjacent to exactly one other cluster leaf-clusters, and Corollary 5.1. Using the network decomposition al- all other clusters inner-clusters. gorithm of [12], in only O(k) rounds, PP and DP Definition 6.1. [9] For a given δ and a positive intecan be approximated by a factor O(n1/k ) w.h.p. For ger k, the cluster tree CTk is recursively defined as k ∈ Θ(log n), this gives a constant factor approximafollows: tion in O(log n) rounds. CT1 := (C1 , A1 ), C1 := {C0 , C1 , C2 , C3 } 6 Lower Bound A1 := {(C0 , C1 ), (C0 , C2 ), (C1 , C3 )} We derive time lower bounds for distributed approxima`(C0 , C1 ) := (δ, δ 2 ), `(C0 , C2 ) := (δ 2 , δ 3 ), bility of packing problems, even in the LOCAL model. `(C1 , C3 ) := (δ, δ 2 ) More precisely, we prove lower bounds for the most basic packing problems, the fractional maximum match- Given CTk−1 , CTk is obtained in two steps: For ing problem (FMM). Our general approach follows [9] each inner-cluster Ci , add a new leaf-cluster C 0 with i in which similar results are obtained for minimum ver- `(Ci , C 0 ) := (δ k+1 , δ k+2 ). For each leaf-cluster Ci with i tex cover which is a covering problem. Specifically, our (Cp , Ci ) ∈ A and `(Cp , Ci ) = (δ p , δ p+1 ), add new packing lower bound graph is structurally similar (al- leaf-clusters C 0 with `(Ci , C 0 ) := (δ j , δ j+1 ) for j = j j though with subtle differences) to the one used in [9]. 1 . . . k + 1, j 6= p + 1. Let Ei denote the set of edges incident to node vi . FMM isP the natural LP relaxation Tk0 be two instances of P of MM and defined Definition 6.2. Let Tk and 0 as max ej ∈E yj , subject to CTk . Further, let Ci and Ci be corresponding clusters in vj ∈Ei yj ≤ 1, ∀vi ∈ V and yj ≥ 0, ∀ej ∈ E. The outcome of an edge’s decision Tk and Tk0 , respectively. We obtain the cluster graph (yj ) in a k-local computation is entirely based on the CGk by adding an arc `(Ci , Ci0 ) := (1, 1) for all clusters information gathered within its k-neighborhood. The Ci ∈ CTk . Further, we define n0 := |C0 ∪ C00 |. This idea for the lower bound is to construct a graph family uniquely defines the size of all clusters. Gk = (V, E) in which, after k rounds of communication, Figure 1 shows CT2 and CG2 . The shaded subtwo adjacent edges see exactly the same graph topology. graphs correspond to CT1 and CG1 , respectively, the Informally speaking, both of them are equally qualified dashed lines represent the links `(Ci , Ci0 ) := (1, 1). Note to join the matching. However, in Gk , taking the wrong decision will be ruinous and yields a suboptimal global that neither CTk nor CGk define the adjacency on the approximation. The construction of Gk is a two step level of nodes. They merely prescribe for each node process. First, the general structure of Gk is defined the number of neighbors in each cluster. We define 0 0 using the concept of a cluster-graph CGk . Secondly, S0 := C0 ∪ C0 and S1 := C1 ∪ C1 . The layer of a we construct an instance of Gk obeying the properties cluster is the distance to C0 in the cluster tree. Tk and Tk0 denote the two cluster trees constituting CGk . imposed by CGk . 4 The primal and dual variables x and y are divided by i j P 1 P minj∈Ni b1 ` aj` x` and maxi∈Nj c ` a`i y` , respectively. j

i

6.2 The Lower Bound Graph Gk Having defined the cluster graph CGk , it is now our goal to obtain

δ4

δ3

δ

δ2

C3 δ2

δ3

δ

δ2

δ4

δ3

C2 δ3

δ4

δ2

δ

δ2

C1 δ3

δ

δ2

Lemma 6.4. When applied to Gk = (V, E) as constructed in Subsection 6.2, any distributed, possibly randomized algorithm which runs for at most k rounds computes, in expectation, a solution of at most ALG ≤ |S0 |/(2δ 2 ) + (|V | − |S0 |).

C0

C0

6.3 Analysis We now derive the lower bounds on the approximation ratio for k-local FMM algorithms. Let OPT be the optimal solution for FMM and let ALG be the solution computed by any algorithm. All nodes in S0 and S1 have the same view and therefore, every edge in E 0 sees the same topology Ve,k .

C’0

Figure 1: Cluster-Tree CT2 and Cluster-Graph CG2 .

a realization of Gk which has the structure imposed by CGk and features the additional property that there are no short cycles. As we must prove that the topologies seen by nodes in S0 and S1 are identical, the absence of short cycles is of great help. Particularly, if there are no cycles of length 2k + 1 and less, all nodes see a tree locally. The girth of a graph G, denoted by g(G), is the length of the shortest cycle in G. Lemma 6.1 states that it is indeed possible to construct Gk as described above. Lemma 6.1. If k + 1 ≤ δ/2, Gk can be constructed such that the following conditions hold: (I) Gk follows the structure of CGk . (II) The girth of Gk is at least g(Gk ) ≥ 2k + 1. 2 (III) Gk has n ≤ 42k δ 4k nodes. Next we show that all nodes in S0 and S1 have the same view and consequently, all edges in E 0 see the same topology. Using the following result from [9] facilitates this task. Lemma 6.2. [9] Let Gk be an instance of a cluster tree CTk with girth g(Gk ) ≥ 2k + 1. The views of all nodes in clusters C0 and C1 are identical up to distance k.

Proof. The fractional value assigned to ei = (u, v) by an algorithm is denoted by yi . The decision of which value yi is assigned to edge ei depends only on the view the topologies Tu,k and Tv,k and the labelings L(Tu,k ) and L(Tv,k ), which ei can collect during the k communication rounds. Assuming that the labeling of Gk is chosen uniformly at random, the labeling L(Tu,k ) for any node u is also chosen uniformly at random. All edges connecting nodes in S0 and S1 see the same topology. If the labels are chosen uniformly at random, it follows that the distribution of the views and therefore the distribution of the yi is the same for all those edges. We call the random variables describing the distribution of the yi , Yi . Let u ∈ S1 be a node of S1 . The node u has δ 2 neighbors in S0 . Therefore, for edges ei between nodes in S0 and S1 , by linearity of expectation, E [Yi ] ≤ 1/δ 2 because otherwise there exist labelings for which the calculated solution is not feasible. By Lemma 6.3, edges ej with both end-points in S0 have the same view as edges between S0 and S1 . Hence, also for the value yj of ej , E [Yj ] ≤ 1/δ 2 must hold. There are |S0 |/2 such edges and therefore the expected total value contributed by edges between two nodes in S0 is at most |S0 |/(2δ 2 ). All edges which do not connect two nodes in S0 , have one end-point in V \ S0 . In order to get a feasible solution, the total value of all edges adjacent to a set of nodes V 0 , can be at most |V 0 |. This can for example be seen by looking at the dual problem, a kind of minimum vertex cover where some edges only have one end node. Taking all nodes of V 0 (assigning 1 to the respective variables) yields a feasible solution for this vertex cover problem. The claim now follows by applying Yao’s minimax principle.

Because Gk has girth at least 2k + 1 by Lemma 6.1, the two cluster-trees Tk and Tk0 constituting Gk must have girth 2k + 1 as well. It follows from Lemma 6.2 that the desired equality of views holds for both Tk and Tk0 . We now derive the lower bound. Lemma 6.4 gives an Based on this fact, it is now easy to show that equality upper bound on the number of nodes chosen by any klocal FMM algorithm. Choosing all edges within S0 is of views holds in Gk , too. feasible, hence, |OPT | ≥ |S0 |/2. In order to establish Lemma 6.3. Let Gk be an instance of a cluster graph a relationship between n, |S0 |, δ, and k, we bound n k+1 ) using a geometric series. The CGk with girth g(Gk ) ≥ 2k + 1. The views of all nodes as n ≤ |S0 |(1 + δ−(k+1) in clusters S0 and S1 are identical up to distance k. second lower bound then follows easily from ∆ = δ k+2 .

Theorem 6.1. For all pairs (n, k) and (∆, k), there are graphs G and a constant c ≥ 1/4, such that in k communication rounds, every distributed algorithm for 2 FMM ¡on G has¢ approximation ratios at least Ω(nc/k /k) 1/k and Ω ∆ /k , respectively. p By setting k = β log n/ log log n and k = β log ∆/ log log ∆, respectively, for a constant β > 0, we obtain the following corollary. Corollary 6.1. In order to obtain a polylogarithmic or constant approximation ratio, every p distributed algorithm for FMM requires at least Ω( log n/ log log n) and Ω(log ∆/ log log ∆) communication rounds. The same lower bounds hold for the construction of maximal matchings and maximal independent sets. Remark: The algorithm in Section 5 achieves a polylogarithmic approximation in O(log ∆/ log log ∆) communication rounds. Therefore, for polylogarithmic approximations, our lower bound for FMM is tight. 7 Conclusions It is interesting to view local computation in a wider context of computational models. Approximation algorithms and online algorithms try to bound the degradation of a globally optimal solution caused by limited computational resources and knowledge about the future, respectively. More recently, the “price of anarchy,” has been proposed to measure the suboptimality resulting from selfish individuals [20]. In a similar spirit, our paper sheds light on the price of locality, i.e., the degradation of a globally optimal solution if each individual’s knowledge is restricted to its neighborhood or local environment. Specifically, the upper and lower bounds presented in this paper characterize the achievable trade-off between local information and the quality of a global solution of covering and packing problems. References [1] N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567–583, 1986. [2] Y. Bartal, J. W. Byers, and D. Raz. Global Optimization Using Local Information with Applications to Flow Control. In Proc. of the 38th Symp. on Foundations of Computer Science (FOCS), pages 303–312, 1997. [3] D. Dubhashi, A. Mei, A. Panconesi, J. Radhakrishnan, and A. Srinivasan. Fast Distributed Algorithms for (Weakly) Connected Dominating Sets and Linear-Size Skeletons. In Proc. of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 717–724, 2003. [4] M. Elkin. Unconditional Lower Bounds on the TimeApproximation Tradeoffs for the Distributed Minimum Spanning Tree Problem. In Proc. of the 36th ACM Symposium on Theory of Computing (STOC), 2004.

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