A Deterministic Algorithm for Maximizing Submodular Functions

The problem of maximizing a non-negative submodular function was introduced by Feige, Mirrokni, and Vondrak [FOCS’07] who provided a deterministic local-search based algorithm that guarantees an approximation ratio of 1 3 , as well as a randomized 2 5 -approximation algorithm. An extensive line of research followed and various algorithms with improving approximation ratios were developed, all of them are randomized. Finally, Buchbinder et al. [FOCS’12] presented a randomized 1 2 -approximation algorithm, which is the best possible. This paper gives the first deterministic algorithm for maximizing a non-negative submodular function that achieves an approximation ratio better than 1 3 . The approximation ratio of our algorithm is 2 5 . Our algorithm is based on recursive composition of solutions obtained by the local search algorithm of Feige et al. We show that the 2 5 approximation ratio can be guaranteed when the recursion depth is 2, and leave open the question of whether the approximation ratio improves as the recursion depth increases.