Metric TSP

Keywords and Synonyms

Metric traveling salesman problem; Metric traveling salesperson problem  

Problem Definition

The Traveling Salesman Problem (TSP) is the following optimization problem:

Input::

A complete loopless undirected graph $G = (V, E, w)$ with a weight function ${ w \colon E \to \mathbb{Q}_{\ge 0} }$ that assigns to each edge a non-negative weight.

Feasible solutions::

All Hamiltonian tours, i. e, the subgraphs H of G that are connected, and each node in them that has degree two.

Objective function::

The weight function $w(H) = \sum_{e \in H} w(e)$ of the tour.

Goal::

Minimization.

The TSP is an ${ \textsf{NP} }$-hard optimization problem. This means that a polynomial time algorithm for the TSP does not exist unless ${ \textsf{P} = \textsf{NP} }$. One way out of this dilemma is provided by approximation algorithms. A polynomial time algorithm for the TSP is called an α-approximation algorithm if the tour H produced by the algorithm fulfills \( { w(H) \le...