Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
SIAM Journal on Discrete Mathematics
Asymptotic packing and the random greedy algorithm
Random Structures & Algorithms
Algorithmic Chernoff-Hoeffding inequalities in integer programming
Random Structures & Algorithms
A survey of approximately optimal solutions to some covering and packing problems
ACM Computing Surveys (CSUR)
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators
SIAM Journal on Discrete Mathematics
Optimal solutions for multi-unit combinatorial auctions: branch and bound heuristics
Proceedings of the 2nd ACM conference on Electronic commerce
Non-approximability results for optimization problems on bounded degree instances
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
On the complexity of approximating k-set packing
Computational Complexity
Inapproximability of hypergraph vertex cover and applications to scheduling problems
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Randomized greedy matching. II
Random Structures & Algorithms
Asymptotic packing via a branching process
Random Structures & Algorithms
Greedy approximation via duality for packing, combinatorial auctions and routing
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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In this paper we analyze the complexity of the maximum cardinality b-matching problem in k-uniform hypergraphs. b-matching is the following problem: for a given b ∈ N and a hypergraph H = (V, E), |V| = n, a subset M ⊆ ε with maximum cardinality is sought so that no vertex is contained in more than b hyperedges of M. The b-matching problem is a prototype of packing integer programs with right-hand side b ≥ 1. It has been studied in combinatorics and optimization. Positive approximation results are known for b ≥ 1 and k-uniform hypergraphs (Krysta 2005) and constant factor approximations for general hypergraphs for b = Ω(log n), (Srinivasan 1999, Srivastav, Stangier 1996, Raghavan, Thompson 1987), but the inapproximability of the problem has been studied only for b = 1 and k-uniform and k-partite hypergraphs (Hazan, Safra, Schwartz 2006). Thus the range from b ∈ [2, log n] is almost unexplored. In this paper we give the first inapproximability result for k-uniform hypergraphs: for every 2 ≤ b ≤ k/ log k there no polynomial-time approximation within any ratio smaller than Ω(k/b log k), unless P = NP. Our result generalizes the result of Hazan, Safra and Schwartz from b = 1 to any fixed 2 ≤ b ≤ k/ log k and k-uniform hypergraphs. But the crucial point is that for the first time we can see how b influences the non-approximability. We consider this result as a necessary step in further understanding the nonapproximability of general packing problems. It is notable that the proof of Hazan, Safra and Schwartz cannot be lifted to b ≥ 2. In fact, some new techniques and ingredients are required; the probabilistic proof of the existence of a hypergraph with "almost" disjoint b-matching where dependent events have to be decoupled (in contrast to Hazan et al.) and the generation of some sparse hypergraph.