Approximation Algorithms for Geometric TSP

Abstract

In the Euclidean traveling salesman problem, we are given n nodes in ℝ² (more generally, in ℝ^d and desire the minimum cost salesman tour for these nodes, where the cost of the edge between nodes (x₁,y₁) and (x₂,y₂) is $\sqrt {(x_1 - x_2 )^2 + (y_1 - y_2 )^2 }$ The decision version of the problem (“Does a tour of cost ≤ C exist?”) is NP-hard [651, 346], but is not known to be in NP because of the use of square roots in computing the edge costs. Specifically, there is no known polynomial-time algorithm that, given integers a₁,a₂,..., a_n, C, can decide if $\sum\nolimits_i {\sqrt {a_i } \leqslant C}$ .

Supported by a David and Lucile Packard Fellowship and NSF grant CCR-0098180