A fast approximation algorithm for the multicovering problem
Discrete Applied Mathematics
Fast approximation algorithms for fractional packing and covering problems
Mathematics of Operations Research
Approximation algorithms for NP-hard problems
Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs
SIAM Journal on Computing
Page replacement for general caching problems
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
Journal of Algorithms
New approaches to covering and packing problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms
Approximating covering integer programs with multiplicity constraints
Discrete Applied Mathematics
Approximation algorithms for covering/packing integer programs
Journal of Computer and System Sciences
Covering Problems with Hard Capacities
SIAM Journal on Computing
Fourier meets möbius: fast subset convolution
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Exact and approximate link scheduling algorithms under the physical interference model
Proceedings of the fifth international workshop on Foundations of mobile computing
Set multi-covering via inclusion-exclusion
Theoretical Computer Science
Set Partitioning via Inclusion-Exclusion
SIAM Journal on Computing
Exact Algorithms for Set Multicover and Multiset Multicover Problems
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
An improved approximation algorithm for vertex cover with hard capacities
Journal of Computer and System Sciences
Algorithms and theory of computation handbook
Approximation schemes for deal splitting and covering integer programs with multiplicity constraints
WAOA'04 Proceedings of the Second international conference on Approximation and Online Algorithms
Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms
Discrete Optimization
Hi-index | 5.23 |
Given a universe N containing n elements and a collection of multisets or sets over N, the multiset multicover (MSMC) problem 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 chosen. First, we show that the MSMC without multiplicity constraints problem can be solved in O^*((b+1)^n|F|) time and polynomial space, where b is the maximum coverage requirement and |F| denotes the total number of given multisets over N. (The O^* notation suppresses a factor polynomial in n.) To our knowledge, this is the first known exact algorithm for the MSMC without multiplicity constraints problem. Second, by combining dynamic programming and the inclusion-exclusion principle, we can exactly solve the SMC without multiplicity constraints problem in O((b+2)^n) time. Compared with two recent results, in [Q.-S. Hua, Y. Wang, D. Yu, F.C.M. Lau, Set multi-covering via inclusion-exclusion, Theoretical Computer Science, 410 (38-40) (2009) 3882-3892] and [J. Nederlof, Inclusion exclusion for hard problems, Master Thesis, Utrecht University, The Netherlands, 2008], respectively, ours is the fastest exact algorithm for the SMC without multiplicity constraints problem. Finally, by directly using dynamic programming, we give the first known exact algorithm for the MSMC or the SMC with multiplicity constraints problem in O((b+1)^n|F|) time and O^*((b+1)^n) space. This algorithm can also be easily adapted as a constructive algorithm for the MSMC without multiplicity constraints problem.