Maximizing Non-Monotone Submodular Functions

• Authors: Uriel Feige;Vahab S. Mirrokni;Jan Vondrak
• Venue: FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2007

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. 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. 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).