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

Abstract

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.KeywordsPolynomial TimeTravel Salesman ProblemPolynomial Time AlgorithmLonge PathPolynomial Time Approximation SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.