Algorithmica

, Volume 74, Issue 2, pp 574–588

Randomized Approximation for the Set Multicover Problem in Hypergraphs

Article

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 $$\delta$$.

Keywords

Combinatorial optimization Approximation algorithms  Randomized algorithm Hypergraphs Set cover and set multicover

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