Covering vectors by spaces: Regular matroids

TL;DR

This paper develops a fixed-parameter tractable algorithm for the Space Cover problem on regular matroids, leveraging Seymour's decomposition theorem, with applications to combinatorial optimization problems like Steiner Forest and Multiway Cut.

Contribution

It introduces a parameterized algorithm for Space Cover on regular matroids, expanding the toolkit for combinatorial problems using matroid theory.

Findings

Algorithm runs in 2^{O(k)}||M||^{O(1)} time for totally unimodular matrices.

Shows fixed-parameter tractability of Space Cover on regular matroids.

Connects the problem to classical problems like Steiner Forest and Multiway Cut.

Abstract

Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on regular matroids. We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, in the Space Cover problem, we are given a matrix M and a subset of its columns T; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains all vectors of T. For graphic matroids this problem is essentially Stainer Forest and for cographic matroids this is a generalization of Multiway Cut. Our main result is the algorithm with running time 2^{O(k)}||M|| ^{O(1)} solving Space Cover in the case when M is a totally unimodular matrix over rationals, where k is the size of F. In other words, we show that on regular matroids the problem is fixed-parameter tractable parameterized by the rank of the covering subspace.