## 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*(log*k*) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for **Subset**
*k*-**Connected Subgraph** when all edges with positive cost have their endnodes in the subset.

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Nutov, Z. Approximating minimum-cost edge-covers of crossing biset-families.
*Combinatorica* **34, **95–114 (2014). https://doi.org/10.1007/s00493-014-2773-4

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### Mathematics Subject Classification (2000)

- 05C40
- 05C85
- 68W25