Improved lower bounds on the approximability of the Traveling Salesman Problem

This paper deals with lower bounds on the approximability of different subproblems of the Traveling Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general (unless P = NP). First of all, we present an improved lower bound for the Traveling Salesman Problem with Triangle Inequality, Δ-TSP for short. Moreover our technique, an extension of the method of Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality, respectively, denoted Δ β -TSP for an appropriate β. In case of the Δ-TSP, we obtain a lower bound of 3813/3812 - e on the polynomial-time approximability (for any small e > 0), compared to the previous bound of 5381/5380 - e in [11] In case of the Δ β -TSP, for the relaxed case (β > 1) we present a lower bound of 3803-10β/3804+8β - e, and for the sharpened triangle inequality (1/2 < β < 1), the lower bound is 7611+10β 2 +5β/7612+8β 2 +4β - e The latter result is of interest especially since it shows that the TSP is APX-hard even if one comes arbitrarily close to the trivial case that all edges have the same cost.

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