A Simpler Self-reduction Algorithm for Matroid Path-width

TL;DR

This paper introduces a simpler fixed-parameter tractable algorithm for constructing optimal path-decompositions of matroids, leveraging a self-reduction approach that builds on existing decision algorithms for matroid path-width.

Contribution

It presents a novel, simpler self-reduction FPT algorithm for constructing matroid path-decompositions, improving upon previous complex methods.

Findings

The new algorithm efficiently constructs optimal path-decompositions.

It reduces the problem to repeated calls to a path-width decision routine.

The approach simplifies the process compared to earlier algorithms.

Abstract

Path-width of matroids naturally generalizes the better known parameter of path-width for graphs, and is NP-hard by a reduction from the graph case. While the term matroid path-width was formally introduced by Geelen-Gerards-Whittle [JCTB 2006] in pure matroid theory, it was soon recognized by Kashyap [SIDMA 2008] that it is the same concept as long-studied so called trellis complexity in coding theory, later named trellis-width, and hence it is an interesting notion also from the algorithmic perspective. It follows from a result of Hlineny [JCTB 2006] that the decision problem, whether a given matroid over a finite field has path-width at most t, is fixed-parameter tractable (FPT) in t, but this result does not give any clue about constructing a path-decomposition. The first constructive and rather complicated FPT algorithm for path-width of matroids over a finite field was given by Jeong-Kim-Oum [SODA 2016]. Here we propose a simpler "self-reduction" FPT algorithm for a path-decomposition. Precisely, we design an efficient routine that constructs an optimal path-decomposition of a matroid by calling any subroutine for testing whether the path-width of a matroid is at most t (such as the aforementioned decision algorithm for matroid path-width).