Minimum Sum Set Cover: Structures and Algorithm

2026-05-20

AI Summary

This study investigates the impact of covering order on the total cost in the min-sum set cover problem and its deviation from the classical set cover size. Focusing on hypergraphs and graphs, the work optimizes vertex ordering to prioritize early edge coverage, integrating combinatorial optimization, parameterized algorithms, and hypergraph analysis. It establishes the first upper bound of τ log₂|E| on the ordered cover cost for general hypergraphs and improves the known bound for graphs to 2τ log₂τ. Furthermore, the paper constructs graph instances demonstrating a lower bound of Ω(τ log τ / log log τ) and presents a fixed-parameter tractable algorithm for bounded-rank hypergraphs.

Abstract

A set cover of a hypergraph $H$ is a set of vertices intersecting every hyperedge. In the minimum sum set cover problem, vertices are selected one by one; each edge pays the position of the first vertex that hits it, and the objective is to minimize the total cost. When $H$ is a graph, this is the minimum sum vertex cover problem. A solution is specified by a set cover $S$ together with an ordering of its vertices. While the classical set cover problem seeks to minimize $|S|$, the minimum sum variant favors covering many edges early and may prefer larger covers. This motivates a natural question: how large can the gap between~$\overrightarrowτ$ and $τ$ be? We prove an upper bound $\overrightarrowτ \le τ\log_{2} \lvert E(H)\rvert$, and show that for any positive~$n$, there exists a hypergraph $H$ on $n + 3$ vertices with $τ=3$ and $\overrightarrowτ=n$. For graphs, we obtain stronger bounds: we prove~$\overrightarrowτ \le 2τ\log_{2} τ$, improving the bound of Liu et al.\ [Theor. Comput. Sci., 2025], and we construct graphs with~$\overrightarrowτ = Ω\left( \frac{τ\log τ}{\log\log τ}\right)$, nearly matching this upper bound. On the algorithmic side, we show that minimum sum set cover is fixed-parameter tractable on bounded-rank hypergraphs, parameterized by~$\overrightarrowτ$, extending the algorithm of Liu et al.\ for graphs (i.e., rank-two hypergraphs).

Problem

• minimum sum set cover
• set cover
• hypergraph
• vertex cover
• covering gap

Innovation

• minimum sum set cover
• hypergraph
• fixed-parameter tractability
• approximation bound
• vertex ordering