## Abstract

Let \(b\in {\mathbb {N}}_{\ge 1}\) and let \({\mathcal {H}}=(V,{\mathcal {E}})\) be a hypergraph with maximum vertex degree \({\varDelta }\) and maximum edge size \(l\). A set \(b\)-multicover in \({\mathcal {H}}\) is a set of edges \(C \subseteq {\mathcal {E}}\) such that every vertex in \(V\) belongs to at least \(b\) edges in \(C\). \({\textsc {set }}\, b\text {-}{\textsc {multicover}}\) is the problem of finding a set \(b\)-multicover of minimum cardinality, and for \(b=1\) it is the fundamental set cover problem. Peleg et al. (Algorithmica 18(1):44–66, 1997) gave a randomized algorithm achieving an approximation ratio of \(\delta \cdot \big (1-\big (\frac{c}{n}\big )^\frac{1}{\delta }\big )\), where \(\delta := {\varDelta }- b + 1\) and \(c>0\) is a constant. As this ratio depends on the instance size \(n\) and tends to \(\delta \) as \(n\) tends to \(\infty \), it remained an open problem whether an approximation ratio of \(\delta \alpha \) with a *constant*
\(\alpha < 1\) can be proved. In fact, the authors conjectured that for any fixed \({\varDelta }\) and \(b\), the problem is not approximable within a ratio smaller than \(\delta \), unless \({\mathcal {P}}={\mathcal {NP}}\). We present a randomized algorithm of hybrid type for \({\textsc {set }}\, b\text {-}{\textsc {multicover}}\), \(b \ge 2\), combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of \(\delta \cdot \left( 1 - \frac{11({\varDelta }- b)}{72l} \right) \) for hypergraphs with maximum edge size \(l \in {\mathcal {O}}\left( \max \big \{(nb)^\frac{1}{5},n^\frac{1}{4}\big \}\right) \). In particular, for all hypergraphs where \(l\) is *constant*, we get an \(\alpha \delta \)-ratio with constant \(\alpha < 1\). Hence the above stated conjecture does not hold for hypergraphs with constant \(l\) and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below \(\delta \).

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## Notes

We may assume that the minimum vertex degree is at least \(b\), because otherwise the problem has no solution.

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El Ouali, M., Munstermann, P. & Srivastav, A. Randomized Approximation for the Set Multicover Problem in Hypergraphs.
*Algorithmica* **74**, 574–588 (2016). https://doi.org/10.1007/s00453-014-9962-9

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DOI: https://doi.org/10.1007/s00453-014-9962-9