An optimal streaming algorithm for non-submodular functions maximization on the integer lattice

Abstract

Submodular optimization problem has been concerned in recent years. The problem of maximizing submodular and non-submodular functions on the integer lattice has received a lot of recent attention. In this paper, we study streaming algorithms for the problem of maximizing a monotone non-submodular functions with cardinality constraint on the integer lattice. For a monotone non-submodular function $f:{\textbf {Z}}^{n}_{+}\rightarrow {\textbf {R}}_{+}$ defined on the integer lattice with diminishing-return (DR) ratio $\gamma$, we present a one pass streaming algorithm that gives a $(1-\frac{1}{2^{\gamma }}-\epsilon )$-approximation, requires at most $O(k\epsilon ^{-1}\log {k/\gamma })$ space and $O(\epsilon ^{-1}\log {k/\gamma }\cdot$ $\log {\Vert {\textbf {B}}\Vert _{\infty }})$ update time per element. We then modify the algorithm and improve the memory complexity to $O(\frac{k}{\gamma \epsilon })$. To the best of our knowledge, this is the first streaming algorithm on the integer lattice for this constrained maximization problem.