The problem of maximizing a non-negative submodular function was introduced by Feige, Mirrokni, and Vondrak [FOCS07] who provided a deterministic local-search based algorithm that guarantees an approximation ratio of $frac 1 3$, as well as a randomiz
ed $frac 2 5$-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. [FOCS12] presented a randomized $frac 1 2$-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 $frac 1 3$. The approximation ratio of our algorithm is $frac 2 5$. Our algorithm is based on recursive composition of solutions obtained by the local search algorithm of Feige et al. We show that the $frac 2 5$ 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.
In this note we study the greedy algorithm for combinatorial auctions with submodular bidders. It is well known that this algorithm provides an approximation ratio of $2$ for every order of the items. We show that if the valuations are vertex cover f
unctions and the order is random then the expected approximation ratio imrpoves to $frac 7 4$.
