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 min{cT x: Tx ≤ b, γT x ≡ r (mod m), x ∊ ℤn} 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. 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