Distributed algorithms for covering, packing and maximum weighted matching

Abstract

This paper gives poly-logarithmic-round, distributed δ-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives poly-logarithmicround, distributed δ-approximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted c-Matching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2-approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms. 1 Background and resultsMany distributed systems are composed of components that can only communicate locally, yet need to achieve a global (system-wide) goal involving many components. are widely studied [12,38,39,42,49,54]. It is of specific interest to see which fundamental combinatorial optimization problems admit efficient distributed algorithms that achieve approximation guarantees that are as good as those of the best centralized algorithms. Research in this spirit includes works on Set Cover (Dominating Set) [26,40,41], capacitated dominating set [37], Capacitated Vertex Cover [16,17], and many other problems. This paper presents distributed approximation algorithms for some fundamental covering and packing problems.The algorithms use the standard synchronous communication model: in each round, nodes can exchange a constant number of messages with neighbors and perform local computation [54]. There is no restriction on message size or local computation. The algorithms are efficient-they finish in a number of rounds that is poly-logarithmic in the network size [42]. Covering problemsConsider optimization problems of the following form: given a non-decreasing, continuous, and submodular 2 cost function c : R n + → R + , and a set of constraints C where each constraint S ∈ C is closed upwards, 3 1 Preliminary versions appeared in [34,35].

Set Cover Problems with Small Neighborhood Covers

“…Their algorithm takes O(f log(1/ ) log(m)) iterations, where f is the frequency parameter. For the specific case of f = 2 (the vertex cover scenario), a parallel procedure for producing 1-maximal solutions is implicit in the work of Koufogiannakis and Young [11]. Their procedure runs in O(log m) iterations.…”

“…In the parallel setting, Khuller et al [10] presented a parallel NC algorithm having approximation ratio of 2 + , for any > 0 (see also [8]). Koufogiannakis and Young [11] presented the first parallel algorithm with approximation ratio of 2. Their algorithm is randomized and runs in RNC.…”

“…Covering integer programs (CIP) generalize the set cover problem. These are well studied in both sequential and distributed settings (see [11,5], and references therein).…”

Distributed Symmetry Breaking in Hypergraphs

“…It is also a natural model for the facility location problem (e.g., [38,39]). The vertex-centric model was considered in, e.g., [23]. Our algorithmic results apply to both models (except the one on maximal matching).…”

Distributed Backup Placement in Networks

“…Using a matching algorithm, [27] presents a polylogarithmic time algorithm for MaxCov with k = 1 which guarantees, for any > 0 and with high probability, a (1 + ) 2−r 1−r -approximation, where r is the maximum ratio between a client demand and a server capacity. A 2-approximation of MaxCov with k = 1 and uniform file sizes can be computed using the matching algorithm of [22]. Regarding MinLoad, we are only aware of distributed algorithms which use large messages (i.e., they run in the local model [28]): in [7] it is shown how to find an approximation of optimal semi-matchings in O(∆ 5 ) rounds under the L2 norm.…”