Non Total-Unimodularity Neutralized Simplicial Complexes

Abstract

Given a simplicial complex K with weights on its simplices and a chain on it, the Optimal Homologous Chain Problem (OHCP) is to find a chain with minimal weight that is homologous (over the integers) to the given chain. The OHCP is NP-complete, but if the boundary matrix of K is totally unimodular (TU), it becomes solvable in polynomial time when modeled as a linear program (LP). We define a condition on the simplicial complex called non total-unimodularity neutralized, or NTU neutralized, which ensures that even when the boundary matrix is not TU, the OHCP LP must contain an integral optimal vertex for every input chain. This condition is a property of K, and is independent of the input chain and the weights on the simplices. This condition is strictly weaker than the boundary matrix being TU. More interestingly, the polytope of the OHCP LP may not be integral under this condition. Still, an integral optimal vertex exists for every right-hand side, i.e., for every input chain. Hence a much larger class of OHCP instances can be solved in polynomial time than previously considered possible. As a special case, we show that 2-complexes with trivial first homology group are guaranteed to be NTU neutralized.