Minimum cut bases in undirected networks

Given an undirected, connected network G=(V,E)G=(V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as another graph theoretic problem closely related to it, namely, the cycle basis problem. We consider two versions of the problem: the unconstrained and the fundamental cut basis problem.For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem, and is thus solvable in strongly polynomial time. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each from a spanning tree TT, is shown to be NP-hard. In this proof, we also show that a tree which induces the minimum fundamental cycle basis is also an optimal solution for the minimum fundamental cut basis problem in unweighted graphs.We present heuristics, integer programming formulations and summarize first experiences with numerical tests.