A 1/2-approximation algorithm for maximizing a non-monotone weak-submodular function on a bounded integer lattice

Abstract

Maximizing non-monotone submodular functions is one of the most important problems in submodular optimization. Let $\mathbf {B}=(B_1, B_2,\ldots , B_n)\in {\mathbb {Z}}_+^n$ be an integer vector and $[\mathbf { B}]=\{(x_1,\dots ,x_n) \in {\mathbb {Z}}_+^n: 0\le x_k \le B_k, \forall 1\le k\le n\}$ be the set of all non-negative integer vectors not greater than $\mathbf {B}$. A function $f:[\mathbf { B}] \rightarrow {\mathbb {R}}$ is said to be weak-submodular if $f(\mathbf {x}+\delta \mathbf {1}_k)-f(\mathbf {x})\ge f(\mathbf {y}+\delta \mathbf {1}_k)-f(\mathbf {y})$ for any $k\in \{1,\dots ,n\}$, any pair of $\mathbf {x}, \mathbf {y}\in [\mathbf { B}]$ such that $\mathbf {x}\le \mathbf {y}$ and $x_k =y_k$, and any $\delta \in {\mathbb {Z}}_+$ satisfying $\mathbf {y}+\delta \mathbf {1}_k\in [\mathbf { B}]$. Here $\mathbf {1}_k$ is the vector with the kth component equal to 1 and each of the others equals to 0. In this paper we consider the problem of maximizing a non-monotone and non-negative weak-submodular function on the bounded integer lattice without any constraint. We present an randomized algorithm with an approximation guarantee $\frac{1}{2}$ for the problem.