On the approximation hardness of dense TSP and other path problems

TSP(1, 2), the Traveling Salesman Problem with distances 1 and 2, is the problem of finding a tour with minimum length in a complete weighted graph where each edge has length 1 or 2. Let d(0) satisfy 0 < d(0) < 1/2. We show that TSP(1, 2) has no PTAS on the set of instances where the minimum degree of the subgraph spanned by the edges with length 1 is bounded below by d(0)n where n is the number of vertices. We also show that LONGEST PATH has no PTAS on the set of instances with minimum degree bounded below by d(0)n for all 0 < d(0) < 1/2. (C) 1999 Published by Elsevier Science B.V. All rights reserved.