A Quantum Algorithm for Finding the Minimum
Gaspers, Serge, Li, Jerry Zirui · arXiv (Cornell University) · 1996
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 2ⁿ n^O(1) time [Andreas Björklund et al., 2009]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140ⁿ) [Kazuya Shimizu and Ryuhei Mori, 2022] and for Set Cover with a small number of sets with running time O(1.7274ⁿ |ℱ|^O(1)) [Andris Ambainis et al., 2019]. 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 set X ⊆ 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 c^|X| n^O(1) time with c 0. Several of our results should be of interest to proponents of classical computing: - We present an inclusion-exclusion algorithm with running time O^*(∑_{i=0}^⌊αn⌋ binom(n,i)), which determines, for each X ⊆ U of size at most α n, 0 ≤ α ≤ 1, whether (X,ℱ) has a k-cover, k-partition, or k-packing. This running time is best-possible, up to polynomial factors. - We prove that for any linear-sized vertex subset X ⊆ V of a graph G = (V,E), the number of minimal dominating sets of G that are subsets of X is O((2-ε)^|X|) for some ε > 0.