Congruency-Constrained TU Problems Beyond the Bimodular Case

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 minc x : T x ≤ b, γ x ≡ r (mod m), x ∈ 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 × 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 using a randomized algorithm. 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.