Bicriteria Algorithms for Maximizing the Difference Between Submodular Function and Linear Function Under Noise

Mengxue Geng, Shufang Gong, Bin Liu, Weili Wu

Abstract: Submodular optimization is an essential problem in many fields due to its diminishing marginal benefit. This property of submodular function plays an important role in many applications. In recent years, the problem of maximizing a non-negative monotone submodular function minus a linear function under various constraints has gradually emerged and is widely used in many practical scenarios such as team formation and recommendation. In this paper, We focus on maximizing a non-negative monotone normalized submodular function minus a linear function under $$\epsilon $$ –multiplicative noise and the result is similar in the case of $$\epsilon $$ -additive noise. Many previous studies were conducted in a noiseless environment, here we consider optimization of this problem in a noisy environment for the first time. In addition, our study will be conducted under two situations, that is, the cardinality constraint and the matroid constraint. Based on these two situations, we propose two bicriteria approximation algorithms respectively and all these algorithms can obtain good results.