Approximation of k-set cover by semi-local optimization
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Approximate set covering in uniform hypergraphs
Journal of Algorithms
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On the hardness of approximating minimization problems
Journal of the ACM (JACM)
Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Using homogeneous weights for approximating the partial cover problem
Journal of Algorithms
Approximation algorithms for maximization problems arising in graph partitioning
Journal of Algorithms
Improved Approximation Algorithms for the Partial Vertex Cover Problem
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
New Constructions of Weak ε-Nets
Discrete & Computational Geometry
Approximation algorithms for partial covering problems
Journal of Algorithms
Algorithmic construction of sets for k-restrictions
ACM Transactions on Algorithms (TALG)
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
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In this paper we present an approximation algorithm for the k-partial vertex cover problem in hypergraphs. Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph with set of vertices V, |V|=n and set of (hyper-)edges $|\mathcal{E}|, |\mathcal{E}| =m$. 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. It is a generalisation of the fundamental (partial) vertex cover problem in graphs and the hitting set problem in hypergraphs. Let l, l≥2 be the maximum size of an edge, Δ be the maximum vertex degree and D be maximum edge degree. For a constant l, l≥2 a non-approximabilty result is known: an approximation ratio better than l cannot be achieved in polynomial-time under the unique games conjecture (Khot and Rageev 2003, 2008). On the other hand, with the primal-dual method (Gandhi, Khuller, Srinivasan 2001) and the local-ratio method (Bar-Yehuda 2001), the l-approximation ratio can be proved. Thus approximations below the l-ratio for large classes of hypergraphs, for example those with constant D or Δ are interesting. In case of graphs (l=2) such results are known. In this paper we break the l-approximation barrier for hypergraph classes with constant D resp. Δ for the partial vertex cover problem in hypergraphs. We propose a randomised algorithm of hybrid type which combines LP-based randomised rounding and greedy repairing. For hypergraphs with arbitrary l, l≥3, and constant D the algorithm achieves an approximation ratio of l(1−Ω(1/(D+1))), and this can be improved to l (1−Ω(1/Δ)) if Δ is constant and k≥m/4. For the class of l-uniform hypergraphs with both l and Δ being constants and l≤4Δ, we get a further improvement to a ratio of $l\left(1-\frac{l-1}{4\Delta}\right)$. The analysis relies on concentration inequalities and combinatorial arguments.