A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid

The Minimal Cycle Basis Problem (MCB) is the following. Given a binary matroid with nonnegative weights assigned to its elements, what is the set of cycles with total smallest weight which generate all of the circuits of the matroid? The answer to this problem also answers in some cases the Sparsest Null Space Basis Problem (NSP) [CP87]. Given a t n matrix A with t < n and rank r, nd a matrix N with the fewest nonzeros whose columns span the null space of A. Coleman and Pothen nd solutions to the latter problem useful in solving the very general Linear Equality Problem: minimize a nonlinear objective function f(x) subject to a matrix equation Ax = b. Many optimization problems are of this form. They are especially concerned when the matrix A is large and sparse. The algorithm given in this paper solves the NSP for totally unimodular matrices, that is matrices in which every square submatrix of A has a determinant of +1, -1 or 0. A matroid is regular if and only if it is representable by the columns of a totally unimodular matrix. Seymour [Sey80] proved that any regular matroid can be decomposed in polynomial time into 1-sums, 2-sums, and 3-sums of graphic matroids, cographic matroids and the special ten element matroid R10. Truemper [Tru90] gives an algorithm which nds such a decomposition in cubic time. An algorithm to solve the MCB problem for graphs is given in [Hor87]. The Gomory-Hu tree of [GH61] solves the MCB problem for cographic matroids. The main technical result of this paper is to show how the minimal cycle