分支定界、分支切割和树搜索算法的通用量子加速
Universal Quantum Speedup for Branch-and-Bound, Branch-and-Cut, and Tree-Search Algorithms
摘要：
Mixed Optimization Programs（MIPs）模拟了计算机科学、运筹学和金融工程中许多令人感兴趣的优化问题。求解MIP一般是NP难的，但有几个求解者已经成功地为中等规模的问题获得了接近最优的解决方案。分支切割算法，结合了分支定界逻辑与切割平面例程，是现代MIP求解器的核心。Montanaro提出了一种量子算法，与经典的分支定界算法相比，在最坏的情况下，当需要每个最优解时，它具有接近二次的加速比。然而，在实践中，接近最优的解决方案是令人满意的，并且通过利用树搜索算法来搜索解决方案树的一部分，经典算法可以比最坏情况保证执行得更好。在本文中，我们提出了一个量子算法，增量量子分支定界，与经典的分支定界算法的每一个输入，即，如果经典分支定界在导致解深度$d$的实例上具有复杂度$Q$，则增量量子分支定界以复杂度$\tilde{O}(\sqrt{Q}d)$提供相同的保证。我们的结果是有效的各种各样的搜索算法，包括基于深度，基于成本，和$A^{\ast}$算法。通用的加速比也得到了分支和切割以及启发式树搜索。我们的算法是直接可比的商业MIP求解器，并保证近二次加速时，$Q \gg d$。我们使用数值模拟来验证$Q \gg d$的典型实例的谢灵顿-柯克帕特里克模型，最大独立集，投资组合优化，以及推断的依赖$Q$的输入大小参数。这使我们能够对这些重要问题的量子算法的典型性能进行预测。
英文摘要：
Mixed Integer Programs (MIPs) model many optimization problems of interest in Computer Science, Operations Research, and Financial Engineering. Solving MIPs is NP-Hard in general, but several solvers have found success in obtaining near-optimal solutions for problems of intermediate size. Branch-and-Cut algorithms, which combine Branch-and-Bound logic with cutting-plane routines, are at the core of modern MIP solvers. Montanaro proposed a quantum algorithm with a near-quadratic speedup compared to classical Branch-and-Bound algorithms in the worst case, when every optimal solution is desired. In practice, however, a near-optimal solution is satisfactory, and by leveraging tree-search heuristics to search only a portion of the solution tree, classical algorithms can perform much better than the worst-case guarantee. In this paper, we propose a quantum algorithm, Incremental-Quantum-Branch-and-Bound, with universal near-quadratic speedup over classical Branch-and-Bound algorithms for every input, i.e., if classical Branch-and-Bound has complexity $Q$ on an instance that leads to solution depth $d$, Incremental-Quantum-Branch-and-Bound offers the same guarantees with a complexity of $\tilde{O}(\sqrt{Q}d)$. Our results are valid for a wide variety of search heuristics, including depth-based, cost-based, and $A^{\ast}$ heuristics. Universal speedups are also obtained for Branch-and-Cut as well as heuristic tree search. Our algorithms are directly comparable to commercial MIP solvers, and guarantee near quadratic speedup whenever $Q \gg d$. We use numerical simulation to verify that $Q \gg d$ for typical instances of the Sherrington-Kirkpatrick model, Maximum Independent Set, and Portfolio Optimization; as well as to extrapolate the dependence of $Q$ on input size parameters. This allows us to project the typical performance of our quantum algorithms for these important problems.
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