Submodular Function Minimization under Covering Constraints

This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2-approximation algorithm for the submodular vertex cover problem based on the half-integrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primal-dual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.

[1]  Fabián A. Chudak,et al.  Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovász extension and non-smooth convex optimization , 2007, SODA '07.

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

[3]  Gagan Goel,et al.  Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions , 2009, FOCS.

[4]  Yurii Nesterov,et al.  Excessive Gap Technique in Nonsmooth Convex Minimization , 2005, SIAM J. Optim..

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

[6]  Chaitanya Swamy,et al.  Network design for information networks , 2005, SODA '05.

[7]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[8]  Jan Vondrák,et al.  Optimal approximation for the submodular welfare problem in the value oracle model , 2008, STOC.

[9]  Joseph Naor,et al.  Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality , 1993, Math. Program..

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

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

[12]  C. SIAMJ..  A FASTER SCALING ALGORITHM FOR MINIMIZING SUBMODULAR FUNCTIONS∗ , 2001 .

[13]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

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

[15]  A. Schrijver.  A Course in Combinatorial Optimization , 1990 .

[16]  Rajeev Kohli,et al.  The Minimum Satisfiability Problem , 1994, SIAM J. Discret. Math..

[17]  U. Feige,et al.  Maximizing Non-monotone Submodular Functions , 2011 .

[18]  Cynthia Dwork,et al.  Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008 , 2008, STOC.

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

[20]  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..

[21]  Michel X. Goemans,et al.  Minimizing submodular functions over families of sets , 1995, Comb..

[22]  A. RÉNY,et al.  ON THE EXISTENCE OF A FACTOR OF DEGREE ONE OF A CONNECTED RANDOM GRAPH , 2004 .

[23]  V. Nagarajan,et al.  MAXIMIZING NON-MONOTONE SUBMODULAR FUNCTIONS UNDER MATROID AND KNAPSACK CONSTRAINTS , 2007 .

[24]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[25]  Harald Niederreiter,et al.  Probability and computing: randomized algorithms and probabilistic analysis , 2006, Math. Comput..

[26]  Lisa Fleischer,et al.  Submodular Approximation: Sampling-based Algorithms and Lower Bounds , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

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

[28]  Satoru Iwata,et al.  A fully combinatorial algorithm for submodular function minimization , 2001, SODA '02.

[29]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[30]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

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

[32]  Satoru Iwata,et al.  A simple combinatorial algorithm for submodular function minimization , 2009, SODA.

[33]  Vahab Mirrokni,et al.  Maximizing Non-Monotone Submodular Functions , 2007, FOCS 2007.

[34]  Béla Bollobás,et al.  Random Graphs , 1985 .

[35]  James B. Orlin,et al.  A faster strongly polynomial time algorithm for submodular function minimization , 2007, Math. Program..

[36]  Vahab S. Mirrokni,et al.  Non-monotone submodular maximization under matroid and knapsack constraints , 2009, STOC '09.

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

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

[39]  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.

[40]  P. Erdos,et al.  On the existence of a factor of degree one of a connected random graph , 1966 .