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

Bin Liu, Zihan Chen, Huijuan Wang, Weili Wu

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 $$f:{\textbf {Z}}^{n}_{+}\rightarrow {\textbf {R}}_{+}$$ f : Z + n → R + defined on the integer lattice with diminishing-return (DR) ratio $$\gamma $$ γ , we present a one pass streaming algorithm that gives a $$(1-\frac{1}{2^{\gamma }}-\epsilon )$$ ( 1 - 1 2 γ - ϵ ) -approximation, requires at most $$O(k\epsilon ^{-1}\log {k/\gamma })$$ O ( k ϵ - 1 log k / γ ) space and $$O(\epsilon ^{-1}\log {k/\gamma }\cdot $$ O ( ϵ - 1 log k / γ · $$\log {\Vert {\textbf {B}}\Vert _{\infty }})$$ log ‖ B ‖ ∞ ) update time per element. We then modify the algorithm and improve the memory complexity to $$O(\frac{k}{\gamma \epsilon })$$ O ( k γ ϵ ) . To the best of our knowledge, this is the first streaming algorithm on the integer lattice for this constrained maximization problem.