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.

中文翻译：

---

双模情况下的一致性约束 TU 问题

整数规划中一个长期存在的悬而未决的问题是具有带界子行列式约束矩阵的整数规划是否可有效求解。其中一个重要的特例是同余约束整数规划 $\min\{c^\top x\colon Tx\leq b, \gamma^\top x\equiv r\pmod*{m}, x\in\mathbb{ Z}^n\}$ 具有完全单模约束矩阵 $T$。此类问题已被证明对于 $m=2$ 是多项式时间可解的，这导致了具有双模约束矩阵的整数程序的有效算法，即满秩矩阵的 $n\times n$ 个子行列式以两个为界绝对值。尽管这些进展严重依赖于已知的具有奇偶约束的组合问题的现有结果，但需要超越双模情况的新方法，即 $m>2$。我们通过几种新技术在这个方向上取得了初步进展。特别是，我们展示了如何使用 $m=3$ 的完全单模约束矩阵有效地确定一致性约束整数程序的可行性。此外，对于一般的 $m$，我们的技术还允许识别不可行问题的平面方向，并推导出问题的解决方案与其松弛之间的接近度的界限。