Abstract :
We analyse an approach to the TSP, introduced by Punnen (Research Report, University of New Brunswick, St. John, Canada, 1996), which is a generalization of approaches by Sarvanov and Doroshko (In: Software: Algorithms and Programs, Math. Inst. Belorusian Acad. Sci., Minsk, No. 31, 1981: 11–13) and Gutin (In: Proceedings of the USSR Conference System Res., Moscow: 1984: 184–185). We show that Punnen’s approach allows one to find the best among View the MathML source tours in the TSP with n cities (n is even) in O(n3) time. We describe an O(n1+β)-time algorithm (for any β∈(0,2]) that constructs the best among View the MathML source tours. This algorithm provides low-complexity solutions to a problem by Burkard et al. (In: Proc. IPCO V, Lecture Notes in Computer Science, Vol. 1084, Berlin: Springer, 1996: 490–504) and may be quite useful for large-scale instances of the TSP. We also show that for every positive integer r there exists an O(r5n)-time algorithm that finds the best among View the MathML source tours. This improves a result of Balas and Simonetti (Management Sci. Res. Report MSRR-617, Canegve Mellon University, Pittsburgh, 1996) who showed that the best among View the MathML source tours can be obtained in time O(r22rn).
