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On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs

✍ Scribed by Cristina Bazgan; Miklos Santha; Zsolt Tuza

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 144 KB
Volume: 31
Category: Article
ISSN: 0196-6774
DOI: 10.1006/jagm.1998.0998

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✦ Synopsis

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 finding a second Hamiltonian cycle when one such cycle is given as part of the input. We investigate the complexity of Ž . approximating this problem where by a feasible solution we mean a nother cycle in the graph, and the quality of the solution is measured by cycle length. First we prove a negative result showing that the Longest Path problem is not constant approximable in cubic Hamiltonian graphs unless P s 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 finding a second cycle in cubic Hamiltonian graphs if a Hamiltonian cycle is given in the input.

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