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48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) (2007)
Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISSN: 0272-5428
ISBN: 0-7695-3010-9
pp: 461-471
ABSTRACT
<p>Submodular maximization generalizes many important problems including Max Cut in directed/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.</p><p> In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative 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 be also achieved by deterministic local search.</p><p> 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, \frac{1}{2} -approximation for symmetric submodular functions is the best one can achieve with a subexponential number of queries. For the case where the function is given explicitly (as a sum of nonnegative submodular functions, each depending only on a constant number of elements), we prove that it is NP-hard to achieve a (\frac{3} {4} + \in ) -approximation in the general case (or a (\frac{5} {6} + \in ) -approximation in the symmetric case).</p>
INDEX TERMS
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CITATION

J. Vondrák, V. S. Mirrokni and U. Feige, "Maximizing Non-Monotone Submodular Functions," 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)(FOCS), Providence, Rhode Island, 2007, pp. 461-471.
doi:10.1109/FOCS.2007.29
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