Euclidean Traveling Salesman Problem

Years and Authors of Summarized Original Work

• 1998; Arora

• 1999; Mitchell

Problem Definition

This entry considers geometric optimization $\mathcal{N}\mathcal{P}$-hard problems like the Euclidean traveling salesman problem and the Euclidean Steiner tree problem. These problems are geometric variants of standard graph optimization problems, and the restriction of the input instances to geometric or Euclidean case arises in numerous applications (see [1, 2]). The main focus of this entry is on the Euclidean traveling salesman problem.

The Euclidean Traveling Salesman Problem (TSP)

For a given set S of n points in the Euclidean space $\mathbb{R}^{d}$, find the minimum length path that visits each point exactly once. The cost δ(x, y) of an edge connecting a pair of points $x,y \in \mathbb{R}^{d}$ is equal to the Euclidean distance between points x and y, that is, $\delta (x,y) = \sqrt{\sum \limits _{i=1 }^{d}}(x_{i} - y_{i})^{2}$, where x = (x₁, …, x _d ) and y = (y₁, …, y_...