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

Abstract

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 $\hat X,\hat Y$ intersect if ^X X ∩ Y ≠ $\not 0$ and cross if both X ∩ Y $\not 0$ and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets $\hat X,\hat Y$ are defined by $\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )$ and $\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )$. A biset-family $\mathcal{F}$ is crossing (intersecting) if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ 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 $\mathcal{F}$ by a minimum-cost set of directed edges. While for intersecting $\mathcal{F}$, a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing $\mathcal{F}$ 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 $\mathcal{F}$ is k-regular if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing $\mathcal{F}$ if in addition both $\mathcal{F}$ and the family of co-bisets of $\mathcal{F}$ are k-regular, our ratios are: $O\left( {\log \frac{{|V|}} {{|V| - k}}} \right)$ if |S ⁺ \ S| = k for all $\hat S \in \mathcal{F}$, and $O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right)$ if |S ⁺ \ S| = k for all $\hat S \in \mathcal{F}$. Using these generic algorithms, we derive for some network design problems the following approximation ratios: $O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right)$ for k-Connected Subgraph, and O(logk) $\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\}$ for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.