Computational complexity of the hamiltonian cycle problem in dense hypergraphs

Authors:
Marek Karpiński;Andrzej Ruciński;Edyta Szymańska
Affiliations:
Department of Computer Science, University of Bonn;Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań;Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

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Abstract

We study the computational complexity of deciding the existence of a Hamiltonian Cycle in some dense classes of k-uniform hypergraphs. Those problems turned out to be, along with the hypergraph Perfect Matching problems, exceedingly hard, and there is a renewed algorithmic interest in them. In this paper we design a polynomial time algorithm for the Hamiltonian Cycle problem for k-uniform hypergraphs with density at least $\tfrac12 + \epsilon$, ε0. In doing so, we depend on a new method of constructing Hamiltonian cycles from (purely) existential statements which could be of independent interest. On the other hand, we establish NP-completeness of that problem for density at least $\tfrac1k - \epsilon$. Our results seem to be the first complexity theoretic results for the Dirac-type dense hypergraph classes.