Spanning circuits in regular 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 Gavenciak, Kral', 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 Gavenciak, Kral', and Oum by showing that for regular matroids• the Minimum Spanning Circuit problem, deciding whether there is a circuit with at most l elements containing T, is FPT parameterized by k = l − |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 l 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.