理解匹配的相关性差距
Understanding the Correlation Gap for Matchings
摘要：
给定匹配多面体中的一组顶点$V$和$|V| = n$，一个权向量$w \in (\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$，和一个概率向量$x \in [0, 1]^{\binom{V}{2}}$，我们研究了数量$\frac{E_{G}[ ν_w(G)]}{\sum_{(u, v) \in \binom{V}{2}} w_{u, v} x_{u, v}}$，其中$G$是一个随机图，其中每个边$e$和权$w_e$以概率$x_e$独立出现，$ν_w(G)$表示$G$的最大匹配权.这个数量是密切相关的差距和竞争解决方案，这是在近似算法，算法博弈论和随机优化设计的重要工具。我们提供了一般和二分图，以及加权和不加权设置的上述数量的下界。最著名的上界是Karp和Sipser的$0.54$，最好的下界是$0.4$。我们证明了对于无权二部图至少是$0.47$，对于加权二部图至少是$0.45$，对于加权一般图至少是$0.43$.为了实现我们的结果，我们构造局部分配计划的对偶，这可能是独立的利益。
英文摘要：
Given a set of vertices $V$ with $|V| = n$, a weight vector $w \in (\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$, and a probability vector $x \in [0, 1]^{\binom{V}{2}}$ in the matching polytope, we study the quantity $\frac{E_{G}[ ν_w(G)]}{\sum_{(u, v) \in \binom{V}{2}} w_{u, v} x_{u, v}}$ where $G$ is a random graph where each edge $e$ with weight $w_e$ appears with probability $x_e$ independently, and let $ν_w(G)$ denotes the weight of the maximum matching of $G$. This quantity is closely related to correlation gap and contention resolution schemes, which are important tools in the design of approximation algorithms, algorithmic game theory, and stochastic optimization. We provide lower bounds for the above quantity for general and bipartite graphs, and for weighted and unweighted settings. he best known upper bound is $0.54$ by Karp and Sipser, and the best lower bound is $0.4$. We show that it is at least $0.47$ for unweighted bipartite graphs, at least $0.45$ for weighted bipartite graphs, and at lea st $0.43$ for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual which may be of independent interest.
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