  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).
