On the Maximum-Weight Basis Problem

TL;DR

This paper characterizes the facets of the bases polytope of a matroid using locked subsets and shows that finding the maximum-weight basis is polynomial-time solvable for certain matroids.

Contribution

It proves that nontrivial facets are described by locked subsets and establishes polynomial solvability for maximum-weight basis in matroids with polynomially many locked subsets.

Findings

Facets of the bases polytope are characterized by locked subsets.

Maximum-weight basis problem is polynomial for certain matroids.

Class of matroids includes uniform and is closed under 2-sums.

Abstract

Let M to be a matroid defined on a finite set E. A subset L of E is locked in M if L is 2-connected in M, the complement of L is 2-connected in the dual M*, and min{r(L), r*(complement of L)} is greater than 1. In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains uniform matroids.