Maximizing Non-monotone Submodular Functions

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.