Primal-Dual Approximation Algorithms for Submodular Vertex Cover Problems with Linear/Submodular Penalties

In this paper, we introduce two variants of the submodular vertex cover problem, namely, the submodular vertex cover problems with linear and submodular penalties, for which we present two primal-dual approximation algorithms with approximation ratios of 2 and 4, respectively. Implementing the primal-dual framework directly on the dual programs of the linear program relaxations for these two variants cannot guarantee the dual ascending process terminates in polynomial time. To overcome this difficulty, we relax the two dual programs to slightly weaker versions which lead to two primal-dual approximation algorithms with the aforeclaimed approximation ratios.

[1]  Dror Rawitz,et al.  An Extension of the Nemhauser--Trotter Theorem to Generalized Vertex Cover with Applications , 2010, SIAM J. Discret. Math..

[2]  Martin Grötschel,et al.  Mathematical Programming The State of the Art, XIth International Symposium on Mathematical Programming, Bonn, Germany, August 23-27, 1982 , 1983, ISMP.

[3]  Yu Li,et al.  Improved Approximation Algorithms for the Facility Location Problems with Linear/submodular Penalty , 2013, COCOON.

[4]  Samir Khuller,et al.  Algorithms for facility location problems with outliers , 2001, SODA '01.

[5]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[6]  Nader H. Bshouty,et al.  Massaging a Linear Programming Solution to Give a 2-Approximation for a Generalization of the Vertex Cover Problem , 1998, STACS.

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

[8]  Valentine Kabanets,et al.  Correlation Bounds and #SAT Algorithms for Small Linear-Size Circuits , 2015, COCOON.

[9]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.

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

[11]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

[12]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[13]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[14]  Dachuan Xu,et al.  A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties , 2011, Algorithmica.

[15]  Dror Rawitz,et al.  On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique , 2005, SIAM J. Discret. Math..

[16]  David P. Williamson,et al.  A note on the prize collecting traveling salesman problem , 1993, Math. Program..

[17]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .

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

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

[20]  Eran Halperin,et al.  Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs , 2000, SODA '00.

[21]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[22]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[23]  Dorit S. Hochbaum,et al.  Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations , 2002, Eur. J. Oper. Res..

[24]  Randeep Bhatia,et al.  Book review: Approximation Algorithms for NP-hard Problems. Edited by Dorit S. Hochbaum (PWS, 1997) , 1998, SIGA.

[25]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[26]  Satoru Iwata,et al.  Submodular Function Minimization under Covering Constraints , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[27]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[28]  Satoru Iwata,et al.  A push-relabel framework for submodular function minimization and applications to parametric optimization , 2003, Discret. Appl. Math..

[29]  George Karakostas,et al.  A better approximation ratio for the vertex cover problem , 2005, TALG.