A Weighted Linear Matroid Parity Algorithm
Satoru Iwata, Yusuke Kobayashi · SIAM Journal on Computing · 2021
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.