An approximation algorithm for the partial vertex cover problem in hypergraphs

Abstract

Let $\mathcal {H}=(V,\mathcal {E})$ be a hypergraph with set of vertices $V, n:=|V|$ and set of (hyper-)edges $\mathcal {E}, m:=|\mathcal {E}|$. Let $l$ be the maximum size of an edge, $\varDelta$ be the maximum vertex degree and $D$ be the maximum edge degree. The $k$-partial vertex cover problem in hypergraphs is the problem of finding a minimum cardinality subset of vertices in which at least $k$ hyperedges are incident. For the case of $k=m$ and constant $l$ it known that an approximation ratio better than $l$ cannot be achieved in polynomial time under the unique games conjecture (UGC) (Khot and Ragev J Comput Syst Sci, 74(3):335–349, 2008), but an $l$-approximation ratio can be proved for arbitrary $k$ (Gandhi et al. J Algorithms, 53(1):55–84, 2004). The open problem in this context has been to give an $\alpha l$-ratio approximation with $\alpha < 1$, as small as possible, for interesting classes of hypergraphs. In this paper we present a randomized polynomial-time approximation algorithm which not only achieves this goal, but whose analysis exhibits approximation phenomena for hypergraphs with $l\ge 3$ not visible in graphs: if $\varDelta$ and $l$ are constant, and $2\le l\le 4\varDelta$, we prove for $l$-uniform hypergraphs a ratio of $l\left( 1-\frac{l-1}{4\varDelta }\right)$, which tends to the optimal ratio 1 as $l\ge 3$ tends to $4\varDelta$. For the larger class of hypergraphs where $l, l \ge 3$, is not constant, but $D$ is a constant, we show a ratio of $l(1-1/6D)$. Finally for hypergraphs with non-constant $l$, but constant $\varDelta$, we get a ratio of $l (1- \frac{2 - \sqrt{3}}{6\varDelta })$ for $k\ge m/4$, leaving open the problem of finding such an approximation for $k < m/4$.