Fractional Linear Matroid Matching is in quasi-NC
Rohit Gurjar, Taihei Oki, Roshan Raj
TL;DR
The paper addresses the problem of placing Fractional Linear Matroid Matching (FLMM) in quasi-NC by linking FLMM to non-commutative Edmonds' problem. It introduces a two-step quasi-NC framework: (i) a parallel construction of an isolating weight set $\\mathcal{W}$ for FLMM polytopes and (ii) a parallel procedure to identify the unique maximum-weight fractional matroid matching via the Pfaffian of the second-order blow-up $A^{\{2\}}$. The key technical contributions include a lattice-based isolating-weight scheme, a refinement of the Pfaffian expansion for rank-two skew-symmetric coefficients, and a deterministic quasi-NC algorithm that attains a perfect FLMM when it exists, along with a corollary for black-box non-commutative Edmonds' with rank-two coefficients. This work advances derandomization in rich matroid-intersection settings and yields a quasi-polynomial hitting-set construction for the restricted non-commutative Edmonds problem, with potential extensions to broader coefficient-rank regimes.
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.