# A Weighted Linear Matroid Parity Algorithm

## TL;DR

This paper introduces a new combinatorial, deterministic polynomial-time algorithm for solving the weighted linear matroid parity problem, extending previous work on unweighted cases using a primal-dual approach.

## Contribution

It presents the first combinatorial, polynomial-time algorithm for the weighted linear matroid parity problem, building on a polynomial matrix formulation and augmenting path techniques.

## Key findings

- Developed a polynomial-time algorithm for weighted linear matroid parity
- Extended combinatorial algorithms from unweighted to weighted cases
- Utilized a primal-dual approach with Pfaffian-based matrix formulation

## Abstract

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, Lov\'asz (1980) 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 (1986) for the unweighted problem.