Lecture Notes in Computer Science Volume 1373, 1998, pp 276-286
Date: 20 Jun 2005

On the approximation of finding A(nother) Hamiltonian cycle in cubic Hamiltonian graphs

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It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian cycles. Finding such a cycle is NP-hard in general, and no polynomial time algorithm is known for the problem of fording a second Hamiltonian cycle when one such cycle is given as part of the input. We investigate the complexity of approximating this problem where by a feasible solution we mean a(nother) cycle in the graph. First we prove a negative result showing that the LONGEST PATH problem is not constant approximable in cubic Hamiltonian graphs unless P = NP. No such negative result was previously known for this problem in Hamiltonian graphs. In strong opposition with this result we show that there is a polynomial time approximation scheme for fording another cycle in cubic Hamiltonian graphs if a Hamiltonian cycle is given in the input.

This research was supported by the ESPRIT Working Group RAND2 n° 21726 and by the bilateral project Balaton, grant numbers 97140 (APAPE, France) and F-36/96 (TéT Alapítvány, Hungary)