Differential approximation results for the traveling salesman problem with distances 1 and 2

Abstract

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any > 0, it is NP-hard to approximate both problems better than within 741/742 + . The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.

Differential Approximation for Some Routing Problems

“…Also TSP(1,2) has no differential approximation scheme, Monnot et al [15] but we cannot deduce immediately that nVRP(1,2) has no differential approximation scheme since wor nV RP and wor T SP can be very far. However, we prove in the following a lower bound for the approximation of nVRP(1,2).…”

Differential Ratio Approximation

“…A very famous restrictive version of min metric tsp is the min tsp12, where edge-weights are all either 1, or 2. In [46], it is proved that this version (as well as, obviously, max tsp12) is approximable within differential ratio 3/4.…”

Differential Ratio Approximation

“…In [46], it is proved that this version (as well as, obviously, max tsp12) is approximable within differential ratio 3/4.…”

“…min set cover, under several assumptions on its worst value, is dealt in [29,39,10], while min weighted set cover is dealt in [24,10]. min tsp and max tsp, as well as, several famous variants of them, min metric tsp, max metric tsp, min tspab (the most famous restrictive case of this problem is min tsp12), max tspab are studied in [45,47,46,39,56]. steiner tree problems under several assumptions on the form of the input-graph and on the edge-weights are dealt in [27].…”