Randomized Approximation for the Set Multicover Problem in Hypergraphs

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 $\delta$.