最大匹配的通信高效核心集

Michael Kapralov, Gilbert Maystre, Jakab Tardos
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引用次数: 2

摘要

本文研究了二部匹配的随机可组合核心集的构造问题。在这个问题中,输入图被随机划分为$k$玩家,每个玩家都向协调器发送一条消息,协调器必须输出输入图中最大匹配的良好近似值。Assadi和Khanna给出了第一个这样的核心集,通过让每个玩家发送一个最大匹配,即每个玩家最多发送n/2个单词,实现了1/9美元的近似值。Bernstein等人将近似因子提高到$1/3$。在本文中,我们展示了Goel, Kapralov和Khanna的匹配骨架结构,这是一个精心选择的(分数)匹配,是一个随机可组合的核心集,它使用每个玩家最多$n-1$个通信单词来实现$1/2-o(1)$近似。我们还显示了由该核心集实现的近似比率的上限为2/3+o(1)$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Communication Efficient Coresets for Maximum Matching
In this paper we revisit the problem of constructing randomized composable coresets for bipartite matching. In this problem the input graph is randomly partitioned across $k$ players, each of which sends a single message to a coordinator, who then must output a good approximation to the maximum matching in the input graph. Assadi and Khanna gave the first such coreset, achieving a $1/9$-approximation by having every player send a maximum matching, i.e. at most $n/2$ words per player. The approximation factor was improved to $1/3$ by Bernstein et al. In this paper, we show that the matching skeleton construction of Goel, Kapralov and Khanna, which is a carefully chosen (fractional) matching, is a randomized composable coreset that achieves a $1/2-o(1)$ approximation using at most $n-1$ words of communication per player. We also show an upper bound of $2/3+o(1)$ on the approximation ratio achieved by this coreset.
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