Nonmonotone Submodular Maximization via a Structural Continuous Greedy Algorithm - (Extended Abstract)

Consider a situation in which one has a suboptimal solution S to a maximization problem which only constitutes a weak approximation to the problem. Suppose that even though the value of S is small compared to an optimal solution OPT to the problem, S happens to be structurally similar to OPT . A natural question to ask in this scenario is whether there is a way of improving the value of S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm which answers this question affirmatively in the context of the Nonmonotone Submodular Maximization Problem. Using this algorithm we are able to improve on the best approximation factor known for this problem. In the Nonmonotone Submodular Maximization Problem we are given a non-negative submodular function f , and the objective is to find a subset maximizing f . This is considered one of the basic submodular optimization problems, generalizing many well known problems such as the Max Cut problem in undirected graphs. The current best approximation factor for this problem is 0.41 given by Gharan and Vondrák. On the other hand, Feige et al. showed that no algorithm can give a 0.5 + ε approximation for it (for any ε > 0). Our method yields an improved 0.42-approximation for the problem.