Computing Girth and Cogirth in Perturbed Graphic Matroids

Abstract

We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids. a cycle of M (A) * . So the cogirth of M (A) is the size of the smallest non-empty cocycle. Again, for this paper, this is the most convenient way to view cogirth.Motivation. The problem of computing the girth of a binary matroid has received a lot of attention due to its well-known connection with coding theory. If A is the parity-check matrix of a binary linear code C, then the distance of C is equal to the girth of the binary matroid M (A). In a landmark paper, Vardy [13] proved that the problem of computing girth in binary matroids is N P-hard. On the other hand, there are significant classes of binary matroids in which one can efficiently compute girth; for example, the class of graphic matroids and the class of cographic matroids. Geelen, Gerards, and Whittle [3] posed the following conjecture. Conjecture 1.3. For any proper minor-closed class M of binary matroids, there is a polynomial-time algorithm for computing the girth of matroids in M.Here a minor-closed class of binary matroids is called proper if it does not, up to isomorphism, contain all binary matroids. In the same paper, Geelen, Gerards, and Whittle announced (without proof) the following result:Theorem 1.4. For each proper minor-closed class M of binary matroids, there exist non-negative integers k and t such that, for each vertically k-connected matroid M ∈ M, there exist matrices A, P ∈ GF(2) r×n such that A is the incidence matrix of a graph, rank(P ) ≤ t, and either M = M (A + P ) or M = M (A + P ) * .In light of this result, our Theorems 1.1 and 1.2 give significant support to Conjecture 1.3; their main shortcomings being: (1) they only apply to sufficiently connected matroids in a minor-closed class, and (2) they only give randomized algorithms.Related work. Barahona and Conforti [1] studied both the girth and cogirth problems for the class of "even-cycle matroids". Let M 1 and M 2 be binary matroids on the same ground set. We call M 1 a rank-t perturbation of M 2 if M 1 has a representation A and M 2 has a representation B such that B − A has rank t. We call M 1 an even-cycle matroid if it is a rank-1 perturbation of a graphic matroid M 2 and r(M 1 ) = r(M 2 ) + 1.Barahona and Conforti gave an efficient deterministic algorithm for computing girth of even-cycle matroids. They also noted that the problem of computing cogirth for these matroids is closely related to the

Submodular Minimization Under Congruency Constraints

“…Notice that Theorem 2 solves the two open questions by Geelen and Kapadia [8] mentioned in the introduction.…”

“…As noted in [8], Conforti and Rao [4] found an efficient deterministic algorithm for the 1-Set Even-Cut Problem. However, even for the 2-Set version, no deterministic procedure is known.…”

“…In particular, (GCCSM) captures the t-Set Even-Cut Problem and t-Set Odd-Cut Problem defined in [8]. There, one is given a constant t, an undirected graph G = (V, E), and sets T 1 , .…”

Submodular Minimization Under Congruency Constraints

“…(i) They ask about a deterministic procedure for the t-Set Even-Cut problem, which they mention as one of the main shortcomings of their approach. As noted in [9], Conforti and Rao [5] found an efficient deterministic algorithm for the 1-Set Even-Cut Problem. However, even for the 2-Set version, no deterministic procedure is known.…”

“…In particular, (GCCSM) captures the t-Set Even-Cut Problem and t-Set Odd-Cut Problem defined in [9]. There, one is given a constant t, an undirected graph G = (V, E), and sets T 1 , .…”