Spanning Circuits in Regular Matroids

TL;DR

This paper advances the algorithmic understanding of spanning circuits in regular matroids, showing fixed-parameter tractability results for certain problems and parameters, while also establishing hardness results that limit these extensions.

Contribution

It extends fixed-parameter tractability results for the Minimum Spanning Circuit and Spanning Circuit problems in regular matroids, and proves hardness results in related classes.

Findings

Minimum Spanning Circuit is FPT parameterized by =- in regular matroids.

Spanning Circuit is FPT parameterized by |T| in regular matroids.

W[1]-hardness of Minimum Spanning Circuit parameterized by  on binary and cographic matroids.

Abstract

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid spanning a set T of given terminal elements. For graphic matroids this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gaven\v{c}iak, Kr\'al', and Oum [ICALP'12], who proved that the case of the problem with |T|=2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gaven\v{c}iak, Kr\'al', and Oum by showing that for regular matroids - the Minimum Spanning Circuit problem, deciding whether there is a circuit with at most \ell elements containing T, is FPT parameterized by k=\ell-|T|; - the Spanning Circuit problem, deciding whether there is a circuit containing T, is FPT parameterized by |T|. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: Minimum Spanning Circuit parameterized by \ell is W[1]-hard on binary matroids even when |T|=1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that Minimum Spanning Circuit parameterized by |T| is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.