Minimum Robust Multi-Submodular Cover for Fairness

TL;DR

This paper introduces MinRF, a new problem for fair multi-submodular coverage, proves its computational hardness, and proposes three bicriteria approximation algorithms with practical applications in information propagation and movie recommendation.

Contribution

The paper formulates MinRF, establishes its inapproximability, and develops three novel bicriteria algorithms for different values of r, demonstrating their effectiveness in real-world scenarios.

Findings

MinRF is hard to approximate within (1-ε)ln m.

Proposed algorithms outperform heuristics in solution quality.

Algorithms are effective in applications like information propagation and recommendations.

Abstract

In this paper, we study a novel problem, Minimum Robust Multi-Submodular Cover for Fairness (MinRF), as follows: given a ground set $V$; $m$ monotone submodular functions $f_1,...,f_m$; $m$ thresholds $T_1,...,T_m$ and a non-negative integer $r$, MinRF asks for the smallest set $S$ such that for all $i \in [m]$, $\min_{|X| \leq r} f_i(S \setminus X) \geq T_i$. We prove that MinRF is inapproximable within $(1-\epsilon)\ln m$; and no algorithm, taking fewer than exponential number of queries in term of $r$, is able to output a feasible set to MinRF with high certainty. Three bicriteria approximation algorithms with performance guarantees are proposed: one for $r=0$, one for $r=1$, and one for general $r$. We further investigate our algorithms' performance in two applications of MinRF, Information Propagation for Multiple Groups and Movie Recommendation for Multiple Users. Our algorithms have shown to outperform baseline heuristics in both solution quality and the number of queries in most cases.