Algorithms for Symmetric Submodular Function Minimization under Hereditary Constraints and Generalizations
Author(s)Goemans, Michel X.; Soto, Jose A.
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We present an efficient algorithm to find nonempty minimizers of a symmetric submodular function f over any family of sets I closed under inclusion. Our algorithm makes O(n[superscript 3]) oracle calls to f and I, where n is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within o(√n/log n) [Z. Svitkina and L. Fleischer, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Washington, DC, 2008, pp. 697--706]. We also present two extensions of the above algorithm. The first extension reports all nontrivial inclusionwise minimal minimizers of f over I using O(n[superscript 3]) oracle calls, and the second reports all extreme subsets of f using O(n[superscript 4]) oracle calls. Our algorithms are similar to a procedure by Nagamochi and Ibaraki [Inform. Process. Lett., 67 (1998), pp. 239--244] that finds all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of size n using O(n[superscript 3]) oracle calls. Their procedure in turn is based on Queyranne's algorithm [M. Queyranne, Math. Program., 82 (1998), pp. 3--12] to minimize a symmetric submodular function by finding pendent pairs. Our results extend to any class of functions for which we can find a pendent pair whose head is not a given element.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
SIAM Journal on Discrete Mathematics
Society for Industrial and Applied Mathematics
Goemans, Michel X., and José A. Soto. “Algorithms for Symmetric Submodular Function Minimization under Hereditary Constraints and Generalizations.” SIAM Journal on Discrete Mathematics 27, no. 2 (April 4, 2013): 1123-1145. © 2013, Society for Industrial and Applied Mathematics
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