A fast algorithm for maximizing a non-monotone DR-submodular integer lattice function

Abstract

In this paper we consider the problem of maximizing a non-monotone and non-negative DR-submodular function on a bounded integer lattice [B→]={(x1,…,xn)∈Z+n:0≤xk≤Bk,∀1≤k≤n} without any constraint, where B→=(B1,…,Bn)∈Z+n. We design an algorithm for the problem and measure its performance by its approximation ratio and the number of value oracle queries it needs, where the latter one is the dominating term in the running time of an algorithm. It has been showed that, for the problem considered, any algorithm achieving an approximation ratio greater than 12 requires an exponential number of value oracle queries. In the literature there are two algorithms that reach 12 approximation guarantee. The first algorithm needs O(n||B||∞) oracle queries. The second one reduces its number of oracle queries to O(nmax⁡{1,log⁡||B→||∞}) but it needs large storage. In this paper we present a randomized approximation algorithm that has an approximation guarantee of 12, calls O(nmax⁡{1,log⁡||B→||∞}) oracle queries and does not need large storage, improving the results of the literature.