加权图的稳定化

摘要：

一个边权图$G=(V,E)$称为稳定图，如果最大权匹配的值等于最大权分数匹配的值。稳定图在一些有趣的博弈论问题中起着重要的作用，如网络讨价还价博弈和合作匹配博弈，因为它们刻画了允许稳定结果的实例。受此启发，在过去的几年里，许多研究人员研究了通过边和顶点删除操作将给定图变成稳定图的算法问题。然而，到目前为止，文献中开发的所有算法结果仅适用于未加权的实例，即，假定$G$的边缘上的单位权重。我们给出了第一个多项式时间算法，找到一个最小的基数子集的顶点，其删除$G$产生一个稳定的图形，任何加权图$G$。该算法是组合的，并利用新的结构性质的基本分数匹配，这是独立的利益。特别是，我们的结果的主要成分之一是多项式时间算法的发展，以计算一个基本的最大重量分数匹配与最小数量的奇循环在其支持。这推广了Balas在30多年前给出的关于未加权匹配的一个基本的经典结果，我们希望在这个特定的应用之外证明它是有用的。与此相反，我们发现的问题，找到一个最小的基数子集的边缘，其删除从加权图$G$产生一个稳定的图形，不承认任何常数因子近似算法，除非$P=NP$。在这种情况下，我们开发了一个$O(Δ)$近似算法的问题，其中$Δ$是一个节点的最大程度的$G$。

英文摘要：

An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of $G$. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from $G$ yields a stable graph, for any weighted graph $G$. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph $G$ yields a stable graph, does not admit any constant-factor approximation algorithm, unless $P=NP$. In this setting, we develop an $O(Δ)$-approximation algorithm for the problem, where $Δ$ is the maximum degree of a node in $G$.

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