We consider box-constrained integer programs with objective $g(Wx) + c^T x$, where $g$ is a "complicated" function with an $m$ dimensional domain. Here we assume we have $n \gg m$ variables and that $W \in \mathbb Z^{m \times n}$ is an integer matrix with coefficients of absolute value at most $\Delta$. We design an algorithm for this problem using only the mild assumption that the objective can be optimized efficiently when all but $m$ variables are fixed, yielding a running time of $n^m(m \Delta)^{O(m^2)}$. Moreover, we can avoid the term $n^m$ in several special cases, in particular when $c = 0$. Our approach can be applied in a variety of settings, generalizing several recent results. An important application are convex objectives of low domain dimension, where we imply a recent result by Hunkenschr\"oder et al. [SIOPT'22] for the 0-1-hypercube and sharp or separable convex $g$, assuming $W$ is given explicitly. By avoiding the direct use of proximity results, which only holds when $g$ is separable or sharp, we match their running time and generalize it for arbitrary convex functions. In the case where the objective is only accessible by an oracle and $W$ is unknown, we further show that their proximity framework can be implemented in $n (m \Delta)^{O(m^2)}$-time instead of $n (m \Delta)^{O(m^3)}$. Lastly, we extend the result by Eisenbrand and Weismantel [SODA'17, TALG'20] for integer programs with few constraints to a mixed-integer linear program setting where integer variables appear in only a small number of different constraints.

翻译：我们考虑带有约束的整数规划问题，其目标函数形式为 $g(Wx) + c^T x$，其中 $g$ 是一个定义在 $m$ 维域上的“复杂”函数。这里假设变量数 $n \gg m$，且 $W \in \mathbb Z^{m \times n}$ 为整数矩阵，其系数的绝对值不超过 $\Delta$。我们设计了一种算法，仅基于一个温和的假设——即当除 $m$ 个变量外的所有变量固定时，目标函数可高效优化——实现了运行时间 $n^m(m \Delta)^{O(m^2)}$。此外，在若干特殊情形下（特别是当 $c = 0$ 时），我们可以避免 $n^m$ 项。该方法可应用于多种场景，并推广了近期多项成果。一个重要应用是低维凸目标函数：当 $W$ 显式给出时，我们改进了 Hunkenschröder 等人 [SIOPT'22] 针对 0-1-超立方体以及 sharp 或可分离凸函数 $g$ 的结果。通过避免直接使用仅适用于可分离或 sharp 函数的邻近性结果，我们不仅匹配了其运行时间，还将推广至任意凸函数。当目标函数仅通过预言机可访问且 $W$ 未知时，我们进一步证明其邻近性框架可在 $n (m \Delta)^{O(m^2)}$ 时间内实现，而非 $n (m \Delta)^{O(m^3)}$。最后，我们将 Eisenbrand 和 Weismantel [SODA'17, TALG'20] 关于少约束整数规划的结果，推广至整数变量仅出现在少量不同约束中的混合整数线性规划情形。