A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover

2026-05-11

AI Summary

This study addresses the Generalized Minimum Sum Set Cover (GMSSC) problem, which seeks a vertex ordering that minimizes the total cover time required to satisfy the coverage demands of all hyperedges. The work proposes an improved linear programming framework coupled with a novel analysis of lower tail bounds for sums of independent Bernoulli random variables to refine the vertex ordering strategy. This approach advances the state-of-the-art approximation ratio for GMSSC from the previous best of 4.642 to 4.509, thereby significantly narrowing the gap toward the theoretical lower bound of 4 under the assumption that P ≠ NP and pushing forward the frontier of approximation algorithms for this fundamental covering problem.

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.

Problem

• generalized min-sum set cover
• hyperedges
• cover time
• vertex ordering
• approximation algorithm

Innovation

• approximation algorithm
• generalized min-sum set cover
• linear programming
• lower-tail bounds
• randomized rounding