Maximizing Non-Monotone Submodular Functions
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Consider a suboptimal solution S for a maximization problem. Suppose S's value is small compared to an optimal solution OPT to the problem, yet S is structurally similar to OPT. A natural question in this setting is whether there is a way of improving S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm, answering this question affirmatively in the setting of the Nonmonotone Submodular Maximization Problem. We 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. Our method yields an 0.42-approximation for this problem, improving on the current best approximation factor of 0.41 given by Gharan and Vondrák [5]. On the other hand, Feige et al. [4] showed a lower bound of 0.5 for this problem.