Integer programs with nearly totally unimodular matrices: the cographic case

It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix M whose subdeterminants are all bounded by a constant in absolute value, can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from M, one obtains a submatrix A that is the transpose of a network matrix. Our approach focuses on the case where A arises from M after removing k rows only, where k is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. First, we derive a strong proximity result for the case where A is a general totally unimodular matrix: Given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of A\; I. Second, for the case where A is transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph G. We observe that if G is 2-connected, then it has no rooted K₂, ₓ-minor for t = (k). We leverage this to obtain a tree-decomposition of G into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming.