A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover
2026-05-11 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study a problem where you want to arrange items so that groups (called hyperedges) are 'covered' as early as possible based on certain coverage needs. They improve on a previous method by finding a way to order items that gets closer to the best possible guarantee. Their technique uses a mathematical programming approach and a new probabilistic analysis involving sums of random variables. This helps better understand the problem and provides a more efficient solution.
generalized min-sum set coverhyperedgesapproximation algorithmlinear programmingLP relaxationcover timeBernoulli random variablesprobabilistic analysisNP-hardnessSODA conference
Authors
Amey Bhangale, Yezhou Zhang
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.