Exact Algorithms for Set Multicover and Multiset Multicover Problems

摘要

Given a universe N containing n elements and a collection of multisets or sets over N, the multiset multicover (MSMC) or the set multicover (SMC) problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets. In this paper, we give various exact algorithms for these two problems, with or without constraints on the number of times a multiset or set may be picked. First, we can exactly solve the MSMC without multiplicity constraints problem in O(((b+1)(c+1))~n) time where b and c (c ≤ b and b ≥ 2) respectively are the maximum coverage requirement and the maximum number of times that each element can appear in a multiset. To our knowledge, this is the first known exact algorithm for the MSMC without multiplicity constraints problem. Second, we can solve the SMC without multiplicity constraints problem in O((b + 2)~n) time. Compared with the two recent results in [Hua et al., Set Multi-covering via inclusion-exclusion, Theoretical Computer Science, 410(38-40):3882-3892 (2009)] and [Nederlof, J.: Inclusion Exclusion for hard problems. Master Thesis. Utrecht University, The Netherlands (2008)], we have given the fastest exact algorithm for the SMC without multiplicity constraints problem. Finally, we give the first known exact algorithm for the MSMC or the SMC with multiplicity constraints problem in O((b + l)~n|F|) time and O((b + 1)~n|F|) space where |F| denotes the total number of given multisets or sets over N.