Capacitated Domination Problem

We consider a generalization of the well-known domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demands of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies the demand of each vertex in V to be met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete (even for its integer version) and provide a polynomial time approximation scheme (PTAS). We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.

[1]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[2]  Mohammad Mahdian,et al.  Approximation Algorithms for Metric Facility Location Problems , 2006, SIAM J. Comput..

[3]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[4]  Deying Li,et al.  A polynomial‐time approximation scheme for the minimum‐connected dominating set in ad hoc wireless networks , 2003, Networks.

[5]  Yuval Rabani,et al.  Approximating k-median with non-uniform capacities , 2005, SODA '05.

[6]  Chaitanya Swamy,et al.  LP-based approximation algorithms for capacitated facility location , 2004, Math. Program..

[7]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

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

[9]  Mohammad Mahdian,et al.  Universal Facility Location , 2003, ESA.

[10]  Judit Bar-Ilan,et al.  Generalized submodular cover problems and applications , 2001, Theor. Comput. Sci..

[11]  Chung-Shou Liao,et al.  K-tuple Domination in Graphs , 2003, Inf. Process. Lett..

[12]  Fabián A. Chudak,et al.  Improved approximation algorithms for a capacitated facility location problem , 1999, SODA '99.

[13]  Éva Tardos,et al.  Approximation algorithms for facility location problems (extended abstract) , 1997, STOC '97.

[14]  Yinyu Ye,et al.  A Multiexchange Local Search Algorithm for the Capacitated Facility Location Problem , 2005, Math. Oper. Res..

[15]  Dorit S. Hochbaum,et al.  Approximation Algorithms for the Set Covering and Vertex Cover Problems , 1982, SIAM J. Comput..

[16]  Chung-Shou Liao,et al.  ALGORITHMIC ASPECT OF k-TUPLE DOMINATION IN GRAPHS , 2002 .

[17]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[18]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[19]  Rajmohan Rajaraman,et al.  Analysis of a local search heuristic for facility location problems , 2000, SODA '98.

[20]  Moses Charikar,et al.  Approximating min-sum k-clustering in metric spaces , 2001, STOC '01.

[21]  Sudipto Guha,et al.  Improved Combinatorial Algorithms for Facility Location Problems , 2005, SIAM J. Comput..

[22]  Samir Khuller,et al.  Capacitated vertex covering , 2003, J. Algorithms.

[23]  Kamesh Munagala,et al.  Local Search Heuristics for k-Median and Facility Location Problems , 2004, SIAM J. Comput..

[24]  David P. Williamson,et al.  Improved approximation algorithms for capacitated facility location problems , 1999, IPCO.

[25]  Reuven Bar-Yehuda,et al.  A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem , 1981, J. Algorithms.

[26]  G Nemhauser,et al.  A multi-exchange local search algorithm for the capacitated facility location problem - (Extended abstract) , 2004 .

[27]  Jiawei Zhang,et al.  Approximation algorithms for facility location problems , 2004 .

[28]  Amin Saberi,et al.  A new greedy approach for facility location problems , 2002, STOC '02.

[29]  Rajiv Gandhi,et al.  An improved approximation algorithm for vertex cover with hard capacities , 2006, J. Comput. Syst. Sci..

[30]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[31]  Sudipto Guha,et al.  A constant-factor approximation algorithm for the k-median problem (extended abstract) , 1999, STOC '99.

[32]  Éva Tardos,et al.  Facility location with nonuniform hard capacities , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[33]  Rajiv Gandhi,et al.  Dependent rounding and its applications to approximation algorithms , 2006, JACM.