On the Minimum Degree Hypergraph Problem with Subset Size Two and the Red-Blue Set Cover Problem with the Consecutive Ones Property

Abstract

Let S be a set and let C _b (blue collection) and C _r (red collection) be two collections of subsets of S. The MDH problem is to find a subset S′ ⊆ S such that S′ ∩ B ≠ ∅ for all B ∈ C _b and |S′ ∩ R| ≤ k for all R ∈ C _r , where k is a given non-negative integer. The RBSC problem is to find a subset S′ ⊆ S with S′ ∩ B ≠ ∅ for all B ∈ C _b which minimizes |{R | R ∈ C _r , S′ ∪ R ≠ ∅ }|. In this paper, improved algorithms are proposed for the MDH problem with k = 1 and all sets in C _b having size two and the RBSC problem with C _b  ∪ C _r having the consecutive ones property. For the first problem, we give an optimal $O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|)$-time algorithm, improving the previous $O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|^{2})$ bound by Dom et al. Our improvement is obtained by presenting a new representation of a dense directed graph, which may be of independent interest. For the second problem, we give an $O(|C_{b}| + |C_{r}| \lg |S| + |S| \lg |S|)$-time algorithm, improving the previous O(|C _b ||S| + |C _r ||S| + |S|²) bound by Chang et al.