Minimum Sum Set Cover: Structures and Algorithm

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~$\overrightarrow{\tau}$ and $\tau$ be? We prove an upper bound $\overrightarrow{\tau} \le \tau \log_{2} \lvert E(H)\rvert$, and show that for any positive~$n$, there exists a hypergraph $H$ on $n + 3$ vertices with $\tau=3$ and $\overrightarrow{\tau}=n$. For graphs, we obtain stronger bounds: we prove~$\overrightarrow{\tau} \le 2\tau \log_{2} \tau$, improving the bound of Liu et al.\ [Theor. Comput. Sci., 2025], and we construct graphs with~$\overrightarrow{\tau} = \Omega\left( \frac{\tau \log \tau}{\log\log \tau}\right)$, 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~$\overrightarrow{\tau}$, extending the algorithm of Liu et al.\ for graphs (i.e., rank-two hypergraphs).