article

Maximizing Non-monotone Submodular Functions

Authors:

Vahab S. Mirrokni,

Jan Vondr�kAuthors Info & Claims

SIAM Journal on Computing, Volume 40, Issue 4

Pages 1133 - 1153

https://doi.org/10.1137/090779346

Published: 01 July 2011 Publication History

Abstract

Submodular maximization generalizes many important problems including Max Cut in directed and undirected graphs and hypergraphs, certain constraint satisfaction problems, and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative (non-monotone) submodular functions. In particular, we give a deterministic local-search $\frac{1}{3}$-approximation and a randomized $\frac{2}{5}$-approximation algorithm for maximizing nonnegative submodular functions. We also show that a uniformly random set gives a $\frac{1}{4}$-approximation. For symmetric submodular functions, we show that a random set gives a $\frac{1}{2}$-approximation, which can also be achieved by deterministic local search. These algorithms work in the value oracle model, where the submodular function is accessible through a black box returning $f(S)$ for a given set $S$. We show that in this model, a $(\frac{1}{2}+\epsilon)$-approximation for symmetric submodular functions would require an exponential number of queries for any fixed $\epsilon>0$. In the model where $f$ is given explicitly (as a sum of nonnegative submodular functions, each depending only on a constant number of elements), we prove NP-hardness of $(\frac{5}{6}+\epsilon)$-approximation in the symmetric case and NP-hardness of $(\frac{3}{4}+\epsilon)$-approximation in the general case.

References

[1]

A. Ageev and M. Sviridenko, An $0.828$ approximation algorithm for uncapacitated facility location problem, Discrete Appl. Math., 93 (1999), pp. 149-156.

[2]

P. Alimonti, Non-oblivious local search for MAX $2$-CCSP with application to MAX DICUT, in Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science, 1997.

[3]

N. Alon and J. H. Spencer, The Probabilistic Method, 2nd ed., Wiley-Interscience, New York, 2000.

[4]

P. Austrin, Improved inapproximability for submodular maximization, in Proceedings of the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization, 2010, pp. 12-24.

[5]

G. Calinescu, C. Chekuri, M. P�l, and J. Vondr�k, Maximizing a submodular set function subject to a matroid constraint, in Proceedings of the 12th International Conference on Integer Programming and Combinatorial Optimization, 2007, pp. 182-196.

[6]

G. Calinescu, C. Chekuri, M. P�l, and J. Vondr�k, Maximizing a submodular function subject to a matroid constraint, SIAM J. Comput., to appear.

[7]

V. Cherenin, Solving some combinatorial problems of optimal planning by the method of successive calculations, in Proceedings of the Conference of Experiences and Perspectives of the Applications of Mathematical Methods and Electronic Computers in Planning, Mimeograph, Novosibirsk, 1962 (in Russian).

[8]

G. Cornuejols, M. Fischer, and G. Nemhauser, Location of bank accounts to optimize float: An analytic study of exact and approximation algorithms, Management Sci., 23 (1977), pp. 789-810.

[9]

G. Cornuejols, M. Fischer, and G. Nemhauser, On the uncapacitated location problem, in Studies in Integer Programming, Ann. Discrete Math. 1, North-Holland, Amsterdam, 1977, pp. 163-177.

[10]

J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, Gordon and Breach, New York, 1970, pp. 69-87.

[11]

U. Feige, A threshold of ${\rm ln}\,n$ for approximating set cover, J. ACM, 45 (1998), pp. 634-652.

[12]

U. Feige, Maximizing social welfare when utility functions are subadditive, in Proceedings of the 38th ACM Symposium on Theory of Computing, 2006, pp. 41-50.

[13]

U. Feige and M. X. Goemans, Approximating the value of two-prover systems, with applications to MAX-$2$SAT and MAX-DICUT, in Proceedings of the 3rd Israel Symposium on Theory and Computing Systems, Tel Aviv, 1995, pp. 182-189.

[14]

U. Feige, V. Mirrokni, and J. Vondr�k, Maximizing non-monotone submodular functions, in Proceedings of the 48th IEEE Symposium on Foundations of Computer Science, 2007, pp. 461-471.

[15]

U. Feige and J. Vondr�k, Approximation algorithms for combinatorial allocation problems: Improving the factor of $1-1/e$, in Proceedings of the 47th IEEE Symposium on Foundations of Computer Science, 2006, pp. 667-676.

[16]

M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, An analysis of approximations for maximizing submodular set functions II, Math. Programming Stud., no. 8 (1978), pp. 73-87.

[17]

L. Fleischer, S. Fujishige, and S. Iwata, A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions, J. ACM, 48 (2001), pp. 761-777.

[18]

A. Frank, Matroids and submodular functions, in Annotated Bibliographies in Combinatorial Optimization, Wiley, New York, 1997, pp. 65-80.

[19]

S. Fujishige, Canonical decompositions of symmetric submodular systems, Discrete Appl. Math., 5 (1983), pp. 175-190.

[20]

M. Goemans, N. Harvey, S. Iwata, and V. Mirrokni, Approximating submodular functions everywhere, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009, pp. 535-544.

[21]

M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.

[22]

B. Goldengorin, G. Sierksma, G. Tijssen, and M. Tso, The data correcting algorithm for the minimization of supermodular functions, Management Sci., 45 (1999), pp. 1539-1551.

[23]

B. Goldengorin, G. Tijssen, and M. Tso, The Maximization of Submodular Functions: Old and New Proofs for the Correctness of the Dichotomy Algorithm, SOM report, University of Groningen, The Netherlands, 1999.

[24]

V. Guruswami, Inapproximability results for set splitting and satisfiability problems with no mixed clauses, Algorithmica, 38 (2004), pp. 451-469.

[25]

V. Guruswami and S. Khot, Hardness of Max $3$-SAT with no mixed clauses, in Proceedings of the 20th IEEE Conference on Computational Complexity, 2005, pp. 154-162.

[26]

E. Halperin and U. Zwick, Combinatorial approximation algorithms for the maximum directed cut problem, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 2001, pp. 1-7.

[27]

J. H�stad, Some optimal inapproximability results, J. ACM, 48 (2001), pp. 798-869.

[28]

V. R. Khachaturov, Mathematical Methods of Regional Programming, Nauka, Moscow, 1989 (in Russian).

[29]

S. Khot, G. Kindler, E. Mossel, and R. O'Donnell, Optimal inapproximability results for MAX-CUT and other two-variable CSPs?, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, 2004, pp. 146-154.

[30]

A. Kulik, H. Shachnai, and T. Tamir, Maximizing submodular functions subject to multiple linear constraints, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009.

[31]

H. Lee, G. Nemhauser, and Y. Wang, Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case, European J. Oper. Res., 94 (1996), pp. 154-166.

[32]

J. Lee, V. Mirrokni, V. Nagarajan, and M. Sviridenko, Non-monotone submodular maximization under matroid and knapsack constraints, in Proceedings of the 41th ACM Symposium on Theory of Computing, 2009, pp. 323-332.

[33]

J. Lee, M. Sviridenko, and J. Vondr�k, Submodular maximization over multiple matroids via generalized exchange properties, in Proceedings of the 12th International Workshop on Approximation Algorithms for Combinatorial Optimization, 2009, pp. 244-257.

[34]

D. Livnat, M. Lewin, and U. Zwick, Improved rounding techniques for the MAX $2$-SAT and MAX DI-CUT problems, in Proceedings of the 9th International Conference on Programming and Combinatorial Optimization, 2002, pp. 67-82.

[35]

L. Lov�sz, Submodular functions and convexity, in Mathematical Programmming: The State of the Art, A. Bachem, M. Gr�tschel, and B. Korte, eds., Springer-Verlag, Berlin, 1983, pp. 235-257.

[36]

M. Minoux, Accelerated greedy algorithms for maximizing submodular set functions, in Optimization Techniques, J. Stoer, ed., Springer-Verlag, Berlin, 1977, pp. 234-243.

[37]

V. Mirrokni, M. Schapira, and J. Vondr�k, Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions, in Proceedings of the ACM Conference on Electronic Commerce, 2008, pp. 70-77.

[38]

E. Mossel, R. O'Donnell, and K. Oleszkiewicz, Noise stability of functions with low influences: Invariance and optimality, in Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, 2005, pp. 21-30.

[39]

G. L. Nemhauser and L. A. Wolsey, Best algorithms for approximating the maximum of a submodular set function, Math. Oper. Res., 3 (1978), pp. 177-188.

[40]

G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, An analysis of approximations for maximizing submodular set functions I, Math. Programming, 14 (1978), pp. 265-294.

[41]

S. Oveis Gharan and J. Vondr�k, Submodular maximization by simulated annealing, in Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011, pp. 1098-1117.

[42]

M. Queyranne, A combinatorial algorithm for minimizing symmetric submodular functions, in Proceedings of the Sixth ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 98-101.

[43]

T. G. Robertazzi and S. C. Schwartz, An accelerated sequential algorithm for producing D-optimal designs, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 341-358.

[44]

A. A. Sch�fer and M. Yannakakis, Simple local search problems that are hard to solve, SIAM J. Comput., 20 (1991), pp. 56-87.

[45]

A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time, J. Combin. Theory Ser. B, 80 (2000), pp. 346-355.

[46]

M. Sviridenko, A note on maximizing a submodular set function subject to a knapsack constraint, Oper. Res. Lett., 32 (2004), pp. 41-43.

[47]

Z. Svitkina and L. Fleischer, Submodular approximation: Sampling-based algorithms and lower bounds, in Proceedings of the 49th IEEE Symposium on Foundations of Computer Science, 2008, pp. 697-706.

[48]

L. Trevisan, Max Cut and the smallest eigenvalue, in Proceedings of the 41st ACM Symposium on Theory of Computing, 2009, pp. 263-272.

[49]

J. Vondr�k, Optimal approximation for the submodular welfare problem in the value oracle model, in Proceedings of the 40th ACM Symposium on Theory of Computing, 2008, pp. 67-74.

[50]

J. Vondr�k, Symmetry and approximability of submodular maximization problems, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, 2009, pp. 651-670.

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Maximizing Non-monotone Submodular Functions
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 40, Issue 4

July 2011

257 pages

ISSN:0097-5397

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 July 2011

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