Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.

翻译：整数线性规划（ILP）是离散优化问题的一种灵活且普遍应用的模型。求解ILP属于\textsf{NP-Hard}问题，但具有重要的实际意义。为ILP获取超二次量子加速一直较为困难，因为处理多约束ILP的经典算法是全局且穷举式的，而能提供超二次加速的量子框架则利用目标函数与可行集的局部结构。我们通过针对Gomory群松弛的量子算法来解决这一问题。ILP的群松弛通过在线性规划（LP）松弛的最优解中舍弃对正值变量的非负性要求，同时保留决策变量的整数性而获得。我们提出了一种具有竞争力的、保持可行性的经典局部搜索算法用于群松弛，并在合理的技术条件下，给出了相应的能实现超二次加速的量子算法。当群松弛满足一种与LP非退化性类似但更强的非退化条件时，我们的方法能得到原始ILP的最优解。否则，群松弛会收紧ILP最优目标值的界限，并通过减小整数间隙来改进后续的分支切割法；我们在多个实际相关的ILP上数值观测到了这一现象。为实现这些结果，我们推导了具有良好谱性质、可高效构造的、用于群松弛的约束保持混合器，这本身也具有独立的研究价值。