A Bicriteria Approximation Algorithm for Minimum Submodular Cost Partial Multi-Cover Problem

Abstract

This paper presents a bicriteria approximation algorithm for the minimum submodular cost partial multi-cover problem (SCPMC), the goal of which is to find a minimum cost sub-collection of sets to fully cover q percentage of total profit of all elements, where the cost on sub-collections is a submodular function, and an element e with covering requirement $r_e$ is fully covered if it belongs to at least $r_e$ picked sets. Such a problem occurs naturally in a social network influence problem.

Assuming that the maximum covering requirement $r_{\max }=\max _e r_e$ is a constant and the cost function is nonnegative, monotone non-decreasing, and submodular, we give the first $(O(b/q\varepsilon ),(1-\varepsilon ))$-bicriteria algorithm for SCPMC, the output of which fully covers at least $(1-\varepsilon )q$-percentage of the total profit of all elements and the performance ratio is $O(b/q\varepsilon )$, where $b=\max _e\left( {\begin{array}{c}f_e\\ r_{e}\end{array}}\right)$ and $f_e$ is the number of sets containing element e. In the case $r\equiv 1$, an $(O(f/q\varepsilon ),1-\varepsilon )$-bicriteria solution can be achieved even when monotonicity requirement is dropped off from the cost function, where f is the maximum number of sets containing a common element.

Supported by NSFC (11771013, 11531011, 61751303) and Major projects of Zhejiang Science Foundation (D19A010003).