A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f:2N→ R⁺, and the objective is to find a subset S⊆ N maximizing $f(S)$. This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by Unconstrained Submodular 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, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133--1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.