A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover

TL;DR

This paper introduces a new approximation algorithm for the generalized min-sum set cover problem, achieving a better approximation ratio of 4.509, which narrows the gap towards the theoretical lower bound.

Contribution

It improves the approximation guarantee for GMSSC to 4.509 using an enhanced LP-based analysis and novel probabilistic bounds.

Findings

Achieved a 4.509-approximation ratio for GMSSC.

Improved upon the previous 4.642 guarantee.

Utilized new lower-tail bounds for Bernoulli sums.

Abstract

We study the \emph{generalized min-sum set cover} (GMSSC) problem, where given a collection of hyperedges $E$ with arbitrary covering requirements $\{k_e \in \mathbb{Z}^+ : e \in E\}$, the objective is to find an ordering of the vertices that minimizes the total cover time of the hyperedges. A hyperedge $e$ is considered covered at the first time when $k_e$ of its vertices appear in the ordering. We present a $4.509$-approximation algorithm for GMSSC, improving upon the previous best-known guarantee of $4.642$~\cite[SODA'21]{BansalBFT21}. Our approach retains the general LP-based framework of Bansal, Batra, Farhadi, and Tetali~\cite{BansalBFT21} but provides an improved analysis that narrows the gap toward the lower bound of $4$-approximation assuming P$\neq$NP. Our analysis takes advantage of the constraints of the linear program in a nontrivial way, along with new lower-tail bounds for the sums of independent Bernoulli random variables, which could be of independent interest.