Abstract
Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/ p( n) tours for some polynomial p( n) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadratic assignment problem, is also considered with respect to the analogous question. Probabilistic, graph-theoretical, group-theoretical and number-theoretical methods and results are used.
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