基于线性规划的无权诱导匹配逼近--突破$Δ$障碍
Linear programming based approximation for unweighted induced matchings --- breaking the $Δ$ barrier
摘要：
如果图的两条边之间没有边连接，则图的匹配是诱导的，而寻找大的诱导匹配是一个非常困难的问题。Lin等人（Approximating weighted induced matchings，Discrete应用数学243（2018）304-310）提供了一种近似算法，该算法具有比率$Δ$，用于最大度$Δ$的图上的诱导匹配问题的加权版本。他们的方法是基于一个整数线性规划制定的完整性差距至少是$Δ-1$，也就是说，他们的方法提供了加权情况下，只有很小的改进空间。然而，对于未加权的情况，我们推测完整性间隙至多为$\frac{5}{8}Δ+O(1)$，并且近似比也可以至少提高到这个值。我们提供了原始-对偶近似算法的比例$(1-ε) Δ+ \frac{1}{2}$一般$Δ$与$ε\approx 0.02005$，和$\frac{7}{3}$的$Δ=3$。此外，我们证明了分数诱导匹配数的阶和最大度的一个最佳可能界。
英文摘要：
A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (Approximating weighted induced matchings, Discrete Applied Mathematics 243 (2018) 304-310) provide an approximation algorithm with ratio $Δ$ for the weighted version of the induced matching problem on graphs of maximum degree $Δ$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $Δ-1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}Δ+O(1)$, and that also the approximation ratio can be improved at least to this value. We provide primal-dual approximation algorithms with ratios $(1-ε) Δ+ \frac{1}{2}$ for general $Δ$ with $ε\approx 0.02005$, and $\frac{7}{3}$ for $Δ=3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.
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