arXiv 2004.087032020-04-18cs.DS

随机加权匹配：$(1-ε)$近似

Stochastic Weighted Matching: $(1-ε)$ Approximation

Soheil Behnezhad, Mahsa Derakhshan

摘要：

设$G=(V, E)$是一个给定的边权图，它的{ \em实现} $\mathcal{G}$是$G$的一个随机子图，它以概率$p$独立地包含每条边$e \in E$.在{ \em随机匹配}问题中，目标是在不知道实现$\mathcal{G}$的情况下，挑选$G$的稀疏子图$Q$，使得$Q$（即图$Q \cap \mathcal{G}$）的已实现边之间的期望最大权匹配近似于整个实现$\mathcal{G}$的最大权匹配。在本文中，我们证明了，对于任何理想的小$ε\in (0, 1)$，每个图$G$有一个子图$Q$，保证$(1-ε)$ -近似，并有最大程度只有$O_{ε, p}(1)$。也就是说，$Q$的最大程度只取决于$ε$和$p$（这两个都是已知的是必要的），而不是例如在$G$中的节点数，边权重等随机匹配问题已被广泛研究的加权和未加权的图形。以前，已知加权图仅存在（接近）半近似子图[Yamaguchi和Maehara，SODA'18; Behnezhad等人，SODA'19]。我们的结果大大改善了这些工作，匹配最先进的未加权图[Behnezhad等人，STOC'20]，并且基本上确定近似因子。

英文摘要：

Let $G=(V, E)$ be a given edge-weighted graph and let its {\em realization} $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e \in E$ independently with probability $p$. In the {\em stochastic matching} problem, the goal is to pick a sparse subgraph $Q$ of $G$ without knowing the realization $\mathcal{G}$, such that the maximum weight matching among the realized edges of $Q$ (i.e. graph $Q \cap \mathcal{G}$) in expectation approximates the maximum weight matching of the whole realization $\mathcal{G}$. In this paper, we prove that for any desirably small $ε\in (0, 1)$, every graph $G$ has a subgraph $Q$ that guarantees a $(1-ε)$-approximation and has maximum degree only $O_{ε, p}(1)$. That is, the maximum degree of $Q$ depends only on $ε$ and $p$ (both of which are known to be necessary) and not for example on the number of nodes in $G$, the edge-weights, etc. The stochastic matching problem has been studied extensively on both weighted and unweighted graphs. Previously, only existence of (close to) half-approximate subgraphs was known for weighted graphs [Yamaguchi and Maehara, SODA'18; Behnezhad et al., SODA'19]. Our result substantially improves over these works, matches the state-of-the-art for unweighted graphs [Behnezhad et al., STOC'20], and essentially settles the approximation factor.

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摘要：

英文摘要：

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