Dynamic programming based algorithms for set multicover and multiset multicover problems

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.

[1]  Stavros G. Kolliopoulos.  Approximating covering integer programs with multiplicity constraints , 2003, Discret. Appl. Math..

[2]  Dorit S. Hochbaum,et al.  Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms , 2006, Discret. Optim..

[3]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[4]  Mikhail J. Atallah,et al.  Algorithms and Theory of Computation Handbook , 2009, Chapman & Hall/CRC Applied Algorithms and Data Structures series.

[5]  Jesper Nederlof,et al.  Inclusion exclusion for hard problems , 2008 .

[6]  Jr. G. Forney,et al.  Viterbi Algorithm , 1973, Encyclopedia of Machine Learning.

[7]  Francis C. M. Lau,et al.  Exact Algorithms for Set Multicover and Multiset Multicover Problems , 2009, ISAAC.

[8]  Stavros G. Kolliopoulos,et al.  Approximation Algorithms for Covering/Packing Integer Programs , 2002, cs/0205030.

[9]  Joseph Naor,et al.  Covering problems with hard capacities , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[10]  Francis C. M. Lau,et al.  Exact and approximate link scheduling algorithms under the physical interference model , 2008, DIALM-POMC '08.

[11]  Francis C. M. Lau,et al.  Set multi-covering via inclusion-exclusion , 2009, Theor. Comput. Sci..

[12]  D. Atkin.  OR scheduling algorithms. , 2000, Anesthesiology.

[13]  Hadas Shachnai,et al.  Approximation schemes for deal splitting and covering integer programs with multiplicity constraints , 2011, Theor. Comput. Sci..

[14]  Sanjeev Khanna,et al.  Page replacement for general caching problems , 1999, SODA '99.

[15]  Evripidis Bampis,et al.  Traffic Grooming in a Passive Star WDM Network , 2004, SIROCCO.

[16]  Guy Kortsarz,et al.  Sum Multicoloring of Graphs , 2000, J. Algorithms.

[17]  D. Hochbaum.  Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems , 1996 .

[18]  Andreas Björklund,et al.  Set Partitioning via Inclusion-Exclusion , 2009, SIAM J. Comput..

[19]  Aravind Srinivasan,et al.  Improved Approximation Guarantees for Packing and Covering Integer Programs , 1999, SIAM J. Comput..

[20]  Éva Tardos,et al.  Fast approximation algorithms for fractional packing and covering problems , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[21]  Andreas Björklund,et al.  Fourier meets möbius: fast subset convolution , 2006, STOC '07.

[22]  Aravind Srinivasan,et al.  New approaches to covering and packing problems , 2001, SODA '01.

[23]  Vijay V. Vazirani,et al.  Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs , 1999, SIAM J. Comput..

[24]  Rajiv Gandhi,et al.  An improved approximation algorithm for vertex cover with hard capacities , 2003, J. Comput. Syst. Sci..

[25]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[26]  David Amzallag,et al.  Capacitated Cell Planning of 4G Cellular Networks , 2007 .

[27]  Joseph Y.-T. Leung,et al.  Scheduling orders for multiple product types with due date related objectives , 2006, Eur. J. Oper. Res..

[28]  Dorit S. Hochbaum,et al.  A fast approximation algorithm for the multicovering problem , 1986, Discret. Appl. Math..