Computational Complexity of the Hamiltonian Cycle Problem in Dense Hypergraphs

Abstract

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 1 2 + , > 0. 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 1 k − . Our results seem to be the first complexity theoretic results for the Dirac-type dense hypergraph classes.

Hamilton cycles in hypergraphs below the Dirac threshold

“…We also consider the tractability of finding a tight Hamilton cycle in a k-graph for k ≥ 3. For such cycles we close the aforementioned 'hardness gap' identified by Karpiński, Ruciński and Szymańska [21] by proving the following theorem. Theorem 1.6.…”

“…That is, we must show that for each 1 ≤ k ≤ r we have A = E(X|j 1 , · · · , j k ), B = 1 νr E(Y |j 1 , · · · , j k ) and C = E(Z|j 1 , · · · , j k ), where A, B and C are the quantities given in Procedure SelectSet. The first two of these equalities were established in [21] (and are straightforward to verify). For the third define f i,j and f i,j as in Procedure SelectSet.…”

“…That is, they showed that for any fixed ε > 0 it is NP-hard to determine whether a graph G with δ(G) ≥ ( 1 2 − ε)|V (G)| admits a Hamilton cycle. More recently, Karpiński, Ruciński and Szymańska [21] gave an analogous result for hypergraphs with minimum codegree below the lower threshold of Theorem 1.1 (actually, their statement pertained only to tight cycles, but the same construction gives the result for -cycles for any < k). Theorem 1.2 (Karpiński, Ruciński and Szymańska [21]).…”

“…More recently, Karpiński, Ruciński and Szymańska [21] gave an analogous result for hypergraphs with minimum codegree below the lower threshold of Theorem 1.1 (actually, their statement pertained only to tight cycles, but the same construction gives the result for -cycles for any < k). Theorem 1.2 (Karpiński, Ruciński and Szymańska [21]). For any 1 ≤ < k and any ε > 0 the following problem is NP-hard: given a k-graph H with δ(H) ≥ ( The same authors observed that for tight cycles this left a 'hardness gap' of ( 1 k , 1 2 ] for the problem of determining whether, for a fixed c in this range, a k-graph H with δ(H) ≥ c|V (H)| admits a tight Hamilton cycle.…”

“…Indeed, it is trivial to determine whether there is a Hamilton -cycle in a k-graph H with minimum degree above the Dirac threshold identified asymptotically in Theorem 1.1 (as by the theorem the answer must be affirmative). Moreover, for tight cycles Karpiński, Ruciński and Szymańska [21] derandomised the proof of Theorem 1.1 to describe a polynomial-time algorithm which actually finds a tight Hamilton cycle in such a k-graph. This raises the question whether the threshold for tractability could lie substantially below the Dirac threshold.…”