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.
For approximate maximum matching problem in the MapReduce model, @cite_18 gave a @math -approximation algorithm, which requires a constant number of rounds and uses @math bits of communication, for any input graph with @math edges.
@cite_18
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The matching problem asks for a large set of disjoint edges in a graph. It is a problem that has received considerable attention in both the sequential and the self-stabilizing literature. Previous work has resulted in self-stabilizing algorithms for computing a maximal (12-approximation) matching in a general graph, as well as computing a 23-approximation on more specific graph types. In this paper, we present the first self-stabilizing algorithm for finding a 23-approximation to the maximum matching problem in a general graph. We show that our new algorithm, when run under a distributed adversarial daemon, stabilizes after at most O(n2) rounds. However, it might still use an exponential number of time steps. We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2Δ-1 colours, and maximal matchings can be computed within O(log* n + Δ) deterministic rounds, where Δ is the maximum degree of the network. We also show how to find maximal independent sets and (Δ + 1)-vertex colourings within O(log* n + Δ2) deterministic rounds. All hidden constants are very small and the algorithms are very simple. We present an O(n log n) algorithm for finding a maximum matching in a circular-arc graph. The best known algorithm for the same problem in general graphs takes O(√|V||E|) time. We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic @math -approximate integral matching algorithm with small polylog worst-case update time. For the @math -approximation regime only a fractional fully-dynamic @math -matching algorithm with worst-case polylog update time was previously known, due to (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio.