An algorithm for weighted fractional matroid matching

TL;DR

This paper presents a polynomial-time algorithm for finding maximum weight fractional matchings in matroids with weighted lines, extending previous work on fractional matchings and their optimization.

Contribution

The paper introduces a new polynomial-time algorithm for maximum weight fractional matroid matchings, generalizing prior results on fractional matchings.

Findings

The algorithm efficiently computes maximum weight fractional matchings.

It extends the polynomial-time solvability to weighted cases.

The method applies to matroids with line sets of rank 1 or 2.

Abstract

Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find for any given weights on the lines in L, a maximum weight fractional matching.