NDLI: Algorithms for Symmetric Submodular Function Minimization under Hereditary Constraints and Generalizations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Goemans, Michel X. Soto, José A.
Copyright Year	2013
Abstract	We present an efficient algorithm to find nonempty minimizers of a symmetric submodular function $f$ over any family of sets ${\cal I}$ closed under inclusion. Our algorithm makes $O(n^3)$ oracle calls to $f$ and ${\cal I}$, 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 $o(\sqrt{n/\log n})$ [Z. Svitkina and L. Fleischer, in Proceedings of the $49$th 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 ${\cal I}$ using $O(n^3)$ oracle calls, and the second reports all extreme subsets of $f$ using $O(n^4)$ 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 $O(n^3)$ 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.
Starting Page	1123
Ending Page	1145
Page Count	23
File Format	PDF
ISSN	08954801
DOI	10.1137/120891502
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-06-25
Access Restriction	Subscribed
Subject Keyword	minimum cut Analysis of algorithms and problem complexity Combinatorial optimization submodular functions symmetric
Content Type	Text
Resource Type	Article
Subject	Mathematics

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