Congruency-Constrained TU Problems Beyond the Bimodular Case

TL;DR

This paper advances the understanding of congruency-constrained integer programming by developing new methods to solve feasibility problems with totally unimodular matrices for moduli greater than two, especially m=3.

Contribution

It introduces novel techniques to decide feasibility for m=3 and to analyze infeasibility and solution proximity for general m in congruency-constrained integer programs.

Findings

Feasibility can be decided efficiently for m=3 with unimodular matrices.

New methods identify flat directions and bounds on solution proximity for general m.

Progress beyond the bimodular case in congruency-constrained integer programming.

Abstract

A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs $\min\{c^\top x\colon\ Tx\leq b,\ \gamma^\top x\equiv r\pmod{m},\ x\in\mathbb{Z}^n\}$ with a totally unimodular constraint matrix $T$. Such problems have been shown to be polynomial-time solvable for $m=2$, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose $n\times n$ subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., for $m>2$. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for $m=3$. Furthermore, for general $m$, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation.