A simple parallel algorithm for the maximal independent set problem
SIAM Journal on Computing
On the parallel complexity of Hamiltonian cycle and matching problem on dense graphs
Journal of Algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Critical chromatic number and the complexity of perfect packings in graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Santa Claus Meets Hypergraph Matchings
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Perfect matchings in large uniform hypergraphs with large minimum collective degree
Journal of Combinatorial Theory Series A
The Complexity of Almost Perfect Matchings in Uniform Hypergraphs with High Codegree
Combinatorial Algorithms
The complexity of vertex coloring problems in uniform hypergraphs with high degree
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Computational complexity of the hamiltonian cycle problem in dense hypergraphs
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Approximating vertex cover in dense hypergraphs
Journal of Discrete Algorithms
Polynomial-time perfect matchings in dense hypergraphs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. Some of these problems are known to be notoriously hard. There is also a renewed interest recently in the very special cases of them. One of the goals of this paper is to shed some light on the tractability barriers for those problems.It has been known that the perfect matching problems are NP-complete for the classes of hypergraphs H with minimum ((k 驴 1) 驴wise) vertex degree 驴 at least c|V(H)| for $c and trivial for $c\ge\frac 12,$ leaving the status of the problems with c in the interval $[\frac 1k,\frac 12)$ widely open. In this paper we show, somehow surprisingly, that $\frac 12$, in fact, is not a threshold for the tractability of the perfect matching problem, and prove the existence of an 驴 0 such that the perfect matching problem for the class of hypergraphs H with 驴 at least $(\frac 12-\epsilon)|V(H)|$ is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest in view of the known results for graphs.