SIAM Journal on Computing, Ahead of Print.
The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovász [Acta Sci. Math., 42 (1980), pp. 121--131] showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. In this paper, we present a combinatorial, deterministic, polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polynomial matrix formulation using Pfaffian and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann [Combinatorica, 6 (1986), pp. 123--150] for the unweighted problem.

中文翻译：

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加权线性拟阵奇偶校验算法

《 SIAM计算杂志》，预印本。
拟阵奇偶校验（或拟阵匹配）问题作为匹配和拟阵交点问题的通用概括而引入，是如此普遍，以至于它需要成倍数量的oracle调用。尽管如此，Lovász[Acta Sci。Math。（42）（1980），第121--131页]表明，该问题允许线性表示拟阵的最小-最大公式和多项式算法。从那时起，已经针对线性拟阵奇偶校验问题开发了有效的算法。在本文中，我们为加权线性拟阵奇偶校验问题提出了一种组合的确定性多项式时间算法。该算法建立在使用Pfaffian的多项式矩阵公式的基础上，并采用基于Gabow和Stallmann的增广路径算法的原始对偶方法[Combinatorica，6（1986），pp。123--150]。