A Simpler Self-reduction Algorithm for Matroid Path-width

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)

Rank-width: Algorithmic and structural results

Rank-width is a width parameter of graphs describing whether it is possible to decompose a graph into a tree-like structure by `simple' cuts. This survey aims to summarize known algorithmic and structural results on rank-width of graphs.Comment: 14 pages; minor revisio