A Simpler Self-reduction Algorithm for Matroid Path-Width

Abstract

The 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 in [J. Geelen, B. Gerards, and G. Whittle, J. Combin. Theory Ser. B, 96 (2006), pp. 405--425] in pure matroid theory, it was soon recognized in [N. Kashyap, SIAM J. Discrete Math., 22 (2008), pp. 256--272] that it is the same concept as the 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 Hliněný [P. Hliněný, J. Combin. Theory Ser. B, 96 (2006), pp. 325--351] 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 in [J. Jeong, E. J. Kim, and S. Oum, in Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2016, pp. 1695--1704]. 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).