Fractional Linear Matroid Matching is in quasi-NC

Abstract: The matching and linear matroid intersection problems are solvable in quasi-NC, meaning that there exist deterministic algorithms that run in polylogarithmic time and use quasi-polynomially many parallel processors. However, such a parallel algorithm is unknown for linear matroid matching, which generalizes both of these problems. In this work, we propose a quasi-NC algorithm for fractional linear matroid matching, which is a relaxation of linear matroid matching and commonly generalizes fractional matching and linear matroid intersection. Our algorithm builds upon the connection of fractional matroid matching to non-commutative Edmonds' problem recently revealed by Oki and Soma~(2023). As a corollary, we also solve black-box non-commutative Edmonds' problem with rank-two skew-symmetric coefficients.