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.

A better differential approximation ratio for symmetric TSP

“…These problems have been tackled several times from a differential approximation point of view. The best ratio obtained so far is 2/3 ( [20,25]), which can be improved up to 3/4 when the weights are restricted to be 1 or 2, [27] (note that in this case the best ratio known for the standard ratio is 7/6, see [28]). Let us also mention that classical optimization strategies have been studied, such as the well known local 2-opt which has been shown in [26] to be a 1/2-differential approximation (while not being a constant standard approximation algorithm even for MinMetricTSP).…”

Differential approximation results for the traveling salesman and related problems

“…Hence, a "good" differential-approximation result implies nothing for the behavior of the approximation algorithm studied when dealing with the standard framework and vice versa. When dealing with maximization problems, we show in [10] that the approximation of a maximization NPO problem Π within differential-approximation ratio δ implies its approximation within standard-approximation ratio δ.…”

Approximation algorithms for some vehicle routing problems

“…Also TSP(1,2) has no differential approximation scheme [22] but we cannot deduce immediately that nVRP (1,2) has no differential approximation scheme since wor nVRP and wor TSP may be very far. However, we prove in the following a lower bound for the differential approximation of nVRP (1,2).…”