A tight linear time (1/2)-approximation for unconstrained submodular maximization

We consider the Unconstrained Sub modular Maximization problem in which we are given a non-negative sub modular function f:2^N → ℝ⁺, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic sub modular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Sub modular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.