Fast algorithms for maximizing monotone nonsubmodular functions

Abstract

In recent years, with the more and more researchers studying the problem of maximizing monotone (nonsubmodular) objective functions, the approximation algorithms for this problem have gotten much progress by using some interesting techniques. In this paper, we develop the approximation algorithms for maximizing a monotone function f with generic submodularity ratio $\gamma$ subject to certain constraints. Our first result is a simple algorithm that gives a $(1-e^{-\gamma } -\epsilon )$-approximation for a cardinality constraint using $O(\frac{n}{\epsilon }log\frac{n}{\epsilon })$ queries to the function value oracle. The second result is a new variant of the continuous greedy algorithm for a matroid constraint. We combine the variant of continuous greedy method with the contention resolution schemes to find a solution with approximation ratio $(\gamma ^2(1-\frac{1}{e})^2-O(\epsilon ))$, and the algorithm makes $O(rn\epsilon ^{-4}log^2\frac{n}{\epsilon })$ queries to the function value oracle.