Approximating minimum-cost edge-covers of crossing biset-families

• Published in: Combinatorica (Budapest. 1981) Vol. 34; no. 1; pp. 95 - 114
• Main Author: Nutov, Zeev
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2014
• Subjects: Combinatorics; Mathematics; Mathematics and Statistics; Original Paper; 05C85; 05C40; 68W25
• ISSN: 0209-9683; 1439-6912
• DOI: 10.1007/s00493-014-2773-4

Part of this paper appeared in the preliminary version [16]. An ordered pair Ŝ = ( S , S + ) of subsets of a groundset V is called a biset if S ⊆ S +; ( V S + ; V S ) is the co-biset of Ŝ . Two bisets intersect if X X ∩ Y ≠ and cross if both X ∩ Y and X + ∪ Y + ≠= V . The intersection and the union of two bisets are defined by and . A biset-family is crossing (intersecting) if for any that cross (intersect). A directed edge covers a biset Ŝ if it goes from S to V S + . We consider the problem of covering a crossing biset-family by a minimum-cost set of directed edges. While for intersecting , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family is k-regular if for any with | V ( X ∪ Y )≥ k +1 that intersect. In this paper we obtain an O (log | V |)-approximation algorithm for arbitrary crossing if in addition both and the family of co-bisets of are k -regular, our ratios are: if | S + \ S | = k for all , and if | S + \ S | = k for all . Using these generic algorithms, we derive for some network design problems the following approximation ratios: for k - Connected Subgraph , and O (log k ) for Subset k - Connected Subgraph when all edges with positive cost have their endnodes in the subset.