We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems, with a variety of applications. The input to the problem is a graph @math that has @math vertices and the set of edges partitioned over @math sites, and an approximation ratio parameter @math . The output is required to be a matching in @math that has to be reported by one of the sites, whose size is at least factor @math of the size of a maximum matching in @math . We show that the communication complexity of this problem is @math information bits. This bound is shown to be tight up to a @math factor, by constructing an algorithm, establishing its correctness, and an upper bound on the communication cost. The lower bound also applies to other graph combinatorial problems in the message-passing communication model, including max-flow and graph sparsification.
The problem of finding an approximate maximum matching in a graph has been studied for various computation models, including the streaming computation model @cite_1 , MapReduce computation model @cite_27 @cite_12 , and a traditional distributed computation model known as @math computation model.
@cite_27 @cite_1 @cite_12
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In this paper we study linear-programming based approaches to the maximum matching problem in the semi-streaming model. In this model edges are presented sequentially, possibly in an adversarial order, and we are only allowed to use a small space. The allowed space is near linear in the number of vertices (and sublinear in the number of edges) of the input graph. The semi-streaming model is relevant in the context of processing of very large graphs. In recent years, there have been several new and exciting results in the semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. In this paper we present several techniques to adapt and optimize linear-programming based approaches in the semi-streaming model. We use the maximum matching problem as a foil to demonstrate the effectiveness of adapting such tools in this model. As a consequence we improve almost all previous results on the semi-streaming maximum matching problem. We also prove new results on interesting variants. We present algorithms for finding large graph matchings in the streaming model. In this model, applicable when dealing with massive graphs, edges are streamed-in in some arbitrary order rather than residing in randomly accessible memory. For e>0, we achieve a @math approximation for maximum cardinality matching and a @math approximation to maximum weighted matching. Both algorithms use a constant number of passes and @math space. In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using O(n) @math n @math 1-1 e @math O(n) @math 1-1 e @math O(n) @math 1-1 e @math O(n) @math 1-e^ -k k^ k-1 (k-1)!=1- 2 k +o(1 k) @math k$ passes in the vertex arrival model using linear space, improving upon previously best known convergence. By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in @math O(Δ) rounds, independently of @math n; here @math Δ is the maximum degree of the graph and @math n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in @math o(Δ) rounds, independently of @math n. Our work gives the first linear-in- @math Δ lower bound for a natural graph problem in the standard @math LOCAL model of distributed computing--prior lower bounds for a wide range of graph problems have been at best logarithmic in @math Δ.