本文提出量子算法，针对集合划分、覆盖和打包问题，在经典枚举算法可二次加速时实现 O*((2+c)^(n/2)) 运行时间，并将图着色改进至 O(1.7956^n)，Domatic Number 达到 O((2-ε)^n)。
摘要：
设 U 为包含 n 个元素的集合，k 为正整数，F 为 U 的（隐式定义的）子集族。我们考虑将 U 划分为 F 中的 k 个集合、用 F 中的 k 个集合覆盖 U、以及将 F 中的 k 个不相交集合打包进 U 的问题。经典上，这些问题可以通过包含-排除法在 O*(2^n) 时间内解决 [BjorklundHK09]。量子上，图着色存在运行时间为 O(1.9140^n) 的更快算法 [ShimizuM22]，对于集合数量较少的 Set Cover 问题，存在运行时间为 O(1.7274^n |F|^O(1)) 的算法 [AmbainisBIKPV19]。在本文中，我们为 Set Partition、Set Cover 和 Set Packing 问题给出了量子加速，前提是存在一个经典枚举算法，该算法适合二次量子加速，即对于 U 的任意子集 X 上的任何子实例，在 O*(c^{|X|}) 时间内（c<2）枚举出限制在（或投影到，对于 k-cover 情况）集合 X 上的 k-划分、k-覆盖或 k-打包（如果存在）的至少一个成员。我们的有界误差量子算法对于 Set Partition、Set Cover 和 Set Packing 的运行时间为 O*((2+c)^(n/2))。当 c<=1.147899 时，我们的算法略快于 O*((2+c)^(n/2))；当 c 接近 1 时，对于 |F| 相对于 n 为次指数的情况，其运行时间与 [AmbainisBIKPV19] 的 Set Cover 算法相匹配。对于图着色问题，我们通过利用小颜色数着色的更快算法来更好地平衡分治步骤，进一步将运行时间改进为 O(1.7956^n)。对于 Domatic Number 问题，我们获得了 O((2-ε)^n) 的运行时间，其中 ε>0。
英文摘要：
Let U be a universe on n elements, let k be a positive integer, and let F be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from F, covering U with k sets from F, and packing k non-intersecting sets from F into U. Classically, these problems can be solved via inclusion-exclusion in O*(2^n) time [BjorklundHK09]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140^n) [ShimizuM22] and for Set Cover with a small number of sets with running time O(1.7274^n |F|^O(1)) [AmbainisBIKPV19]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a subset X of U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in O*(c^{|X|}) time with c<2.
Our bounded-error quantum algorithm runs in O*((2+c)^(n/2)) for Set Partition, Set Cover, and Set Packing. When c<=1.147899, our algorithm is slightly faster than O*((2+c)^(n/2)); when c approaches 1, it matches the running time of [AmbainisBIKPV19] for Set Cover when |F| is subexponential in n.
For Graph Coloring, we further improve the running time to O(1.7956^n) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number, we obtain a O((2-ε)^n) running time for some ε>0.
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