Improved Approximation Lower Bounds for TSP with Distances One and Two

The metric travelling salesman problem Δ-Tsp is the traveling salesman problem in which the distances among cities satisfy the triangle inequality. In this paper, we consider the matric traveling salesman problem Δ(1,2)-Tsp with distances one and two and Δ(1,2,3)-Tsp with distances one, two, and three as the special cases of Δ-Tsp. Since Δ(1,2)-Tsp is NP-complete, it is NP-hard to find an optimal solution for Δ(1,2)-Tsp. So in polynomial time, we with to find an approximate solution for Δ(1,2)-Tsp. owever Δ(1,2)-Tsp is APX-complete, there is a nontrivial approximation lower bound for Δ(1,2)-Tsp. For any ε>0, Engebretsen showed that it is NP-hard to approximate the symmetric Δ(1,2)-Tsp within 5381/5380-ε; the asymmetric Δ(1,2)-Tsp within 2805/2804-ε, and Bochenhauer, et al. showed that it is NP-hard to approximate the symmetric Δ(1,2,3)-Tsp within 3813/3812-ε. In this paper, we improve those lower bounds and show that for any ε>0, it is NP-hard to approximate the symmetric Δ(1,2)-Tsp within 1027/1026-ε (Corollary 4.5); the asymmetric Δ(1,2)-Tsp within 535/534-ε (Corollary 4.7); the symmetric Δ(1,2,3)-Tsp within 817/816-ε (Theorem 5.2); the asymmetric Δ(1,2,3)-Tsp within 364/363-ε (Theorem 5.3). We also show that for any ε>0, it is NP-hard to approximate Shortest-Superstring within 1279/1278-ε (Corollary 6.3)