On the Parameterized Complexity of Girth and Connectivity Problems on Linear Matroids

Abstract

Computing the minimum distance of a linear code is a fundamental problem in coding theory. This problem is a special case of the Matroid Girth problem, where the objective is to compute the length of a shortest circuit in a given matroid. A closely related problem on matroids is the Matroid Connectivity problem where the objective is to compute the connectivity of a given matroid. Given a matroid $M=(E,\mathcal{I})$, a k-separation of M is a partition (X, Y) of E such that $|X|\ge k,\;|Y|\ge k$ and $r(X)+r(Y)-r(E)\le k-1$, where r is the rank function. The connectivity of a matroid M is the smallest k such that M has a k-separation.

In this paper we study the parameterized complexity of Matroid Girth and Matroid Connectivity on linear matroids representable over a field ${\mathbb F_q}$. We consider the parameters–(i) solution size, k, (ii) $\text{ rank }(M)$, and (iii) $\text{ rank }(M)$+q, where M is the input matroid.

We prove that Matroid Girth and Matroid Connectivity when parameterized by $\text{ rank }(M)$, hence by solution size, k, are not expected to have FPT algorithms under standard complexity hypotheses. We then design fast FPT algorithms for Matroid Girth and Matroid Connectivity when parameterized by $\text{ rank }(M)$+q. Finally, since the field size of the linear representation of transversal matroids and gammoids are large we also study Matroid Girth on these specific matroids and give algorithms whose running times do not depend exponentially on the field size.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 267959.