# Integer programs with nearly totally unimodular matrices: the cographic case

arXiv: https://arxiv.org/abs/2407.09477 · HTML: https://mlsanity.com/paper/2407.09477/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [Arithmetic Circuits and Neural Networks for Regular Matroids](https://mlsanity.com/paper/2511.02406/) (2025)

## Abstract (arXiv)

It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix M whose subdeterminants are all bounded by a constant Delta 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,t-minor for t = Omega(k Delta). 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.