Spanning Circuits in Regular Matroids

Abstract

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid containing 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 Gavenčiak, Král’, 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 Gavenčiak, Král’, and Oum by showing that for regular matroids • the M inimum S panning C ircuit problem, deciding whether there is a circuit with at most ℓ elements containing T , is FPT parameterized by k = ℓ − ∣T∣ • the S panning C ircuit 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: M inimum S panning C ircuit 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 M inimum S panning C ircuit parameterized by ∣T∣ is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.