Fast Algorithms for Maximizing Monotone Nonsubmodular Functions

Abstract

Recently, there has been much progress on improving approximation for problems of maximizing monotone (nonsubmodular) objective functions, and many interesting techniques have been developed to solve these problems. In this paper, we develop 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 and 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.

This work was supported in part by the National Natural Science Foundation of China (11971447, 11871442), and the Fundamental Research Funds for the Central Universities.