Randomized Approximation for the Set Multicover Problem in Hypergraphs

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Let $$b\in {\mathbb {N}}_{\ge 1}$$b?N?1 and let $${\mathcal {H}}=(V,{\mathcal {E}})$$H=(V,E) be a hypergraph with maximum vertex degree $${\varDelta }$$Δ and maximum edge size $$l$$l. A set $$b$$b-multicover in $${\mathcal {H}}$$H is a set of edges $$C \subseteq {\mathcal {E}}$$C⊆E such that every vertex in $$V$$V belongs to at least $$b$$b edges in $$C$$C. $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER is the problem of finding a set $$b$$b-multicover of minimum cardinality, and for $$b=1$$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 $$\delta \cdot \big (1-\big (\frac{c}{n}\big )^\frac{1}{\delta }\big )$$?·(1-(cn)1?), where $$\delta := {\varDelta }- b + 1$$?:=Δ-b+1 and $$c>0$$c>0 is a constant. As this ratio depends on the instance size $$n$$n and tends to $$\delta $$? as $$n$$n tends to $$\infty $$?, it remained an open problem whether an approximation ratio of $$\delta \alpha $$?? with a constant$$\alpha < 1$$?<1 can be proved. In fact, the authors conjectured that for any fixed $${\varDelta }$$Δ and $$b$$b, the problem is not approximable within a ratio smaller than $$\delta $$?, unless $${\mathcal {P}}={\mathcal {NP}}$$P=NP. We present a randomized algorithm of hybrid type for $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER, $$b \ge 2$$b?2, combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of $$\delta \cdot \left( 1 - \frac{11({\varDelta }- b)}{72l} \right) $$?·1-11(Δ-b)72l for hypergraphs with maximum edge size $$l \in {\mathcal {O}}\left( \max \big \{(nb)^\frac{1}{5},n^\frac{1}{4}\big \}\right) $$l?Omax{(nb)15,n14}. In particular, for all hypergraphs where $$l$$l is constant, we get an $$\alpha \delta $$??-ratio with constant $$\alpha < 1$$?<1. Hence the above stated conjecture does not hold for hypergraphs with constant $$l$$l and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below $$\delta $$?.