Integer Linear Programming with $n$ binary variables and $m$ many $0/1$-constraints can be solved in time $2^{\tilde O(m^2)} \text{poly}(n)$ and it is open whether the dependence on $m$ is optimal. Several seemingly unrelated problems, which include variants of Closest String, Discrepancy Minimization, Set Cover, and Set Packing, can be modelled as Integer Linear Programming with $0/1$ constraints to obtain algorithms with the same running time for a natural parameter $m$ in each of the problems. Our main result establishes through fine-grained reductions that these problems are equivalent, meaning that a $2^{O(m^{2-\varepsilon})} \text{poly}(n)$ algorithm with $\varepsilon > 0$ for one of them implies such an algorithm for all of them. In the setting above, one can alternatively obtain an $n^{O(m)}$ time algorithm for Integer Linear Programming using a straightforward dynamic programming approach, which can be more efficient if $n$ is relatively small (e.g., subexponential in $m$). We show that this can be improved to ${n'}^{O(m)} + O(nm)$, where $n'$ is the number of distinct (i.e., non-symmetric) variables. This dominates both of the aforementioned running times.

翻译：对于具有 $n$ 个二元变量和 $m$ 个 $0/1$ 约束的整数线性规划问题，其求解时间复杂度为 $2^{\tilde O(m^2)} \text{poly}(n)$，而关于 $m$ 的依赖是否最优仍是开放问题。若干看似无关的问题——包括最近字符串变体、差异最小化、集合覆盖和集合包装——均可建模为带 $0/1$ 约束的整数线性规划问题，从而针对各问题中的自然参数 $m$ 获得具有相同时间复杂度的算法。我们的核心结果通过细粒度归约证明这些问题具有等价性：若对其中任一问题存在 $\varepsilon > 0$ 的 $2^{O(m^{2-\varepsilon})} \text{poly}(n)$ 算法，则所有问题均存在此类算法。在上述设定中，亦可采用直接的动态规划方法获得 $n^{O(m)}$ 时间的整数线性规划算法，当 $n$ 相对较小（例如 $m$ 的亚指数规模）时可能更高效。我们证明该结果可改进为 ${n'}^{O(m)} + O(nm)$，其中 $n'$ 表示互异（即非对称）变量的数量。该时间复杂度优于前述两种运行时间。