There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $\Delta\in \mathbb{Z}_{>0}$, $\Delta$-modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in \mathbb{Z}^{m\times n}$ has full column rank and all $n\times n$ minors of $A$ are within $\{-\Delta, \dots, \Delta\}$. Previous progress on this question, in particular for $\Delta=2$, relies on algorithms that solve an important special case, namely strictly $\Delta$-modular IPs, which further restrict the $n\times n$ minors of $A$ to be within $\{-\Delta, 0, \Delta\}$. Even for $\Delta=2$, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $\Delta$-modular IPs. Prior advances were restricted to prime $\Delta$, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $\Delta$-modular IPs in strongly polynomial time if $\Delta\leq4$.

翻译：近期，关于约束矩阵 $A$ 具有有界子式的整数规划（IPs）$\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ 取得了重要进展。这一研究源于一个著名猜想：对于任意常数 $\Delta\in \mathbb{Z}_{>0}$，$\Delta$-模整数规划（即约束矩阵 $A\in \mathbb{Z}^{m\times n}$ 列满秩，且所有 $n\times n$ 子式均属于 $\{-\Delta, \dots, \Delta\}$）可在多项式时间内高效求解。此前针对该问题（尤其是 $\Delta=2$ 情形）的进展，依赖于求解其重要特例——严格 $\Delta$-模整数规划的算法，其中进一步限制 $A$ 的 $n\times n$ 子式属于 $\{-\Delta, 0, \Delta\}$。即使对于 $\Delta=2$，此类问题仍涵盖最小奇/偶割问题等知名组合优化问题。该猜想即使在严格 $\Delta$-模整数规划情形下仍未解决。先前的进展局限于素数 $\Delta$，从而可应用强大的数论结果。本文首次突破非素数情形，提出不依赖此类强数论素数结果的技术。特别地，我们的方法表明：当 $\Delta\leq4$ 时，存在随机化算法可在强多项式时间内判定严格 $\Delta$-模整数规划的可行性。