The Approximation Ratio of the Greedy Algorithm for the Metric Traveling Salesman Problem

TL;DR

This paper establishes that the greedy algorithm's approximation ratio for the metric Traveling Salesman Problem is logarithmic in the number of nodes, and this result extends to various TSP variants and heuristics.

Contribution

It proves the approximation ratio of the greedy algorithm for the metric TSP is Θ(log n), extending to graphic, Euclidean, rectilinear instances and analyzing the Clarke-Wright heuristic.

Findings

Greedy algorithm has a Θ(log n) approximation ratio for metric TSP.

The same ratio applies to graphic, Euclidean, and rectilinear TSP instances.

Clarke-Wright heuristic also achieves a Θ(log n) approximation ratio.

Abstract

We prove that the approximation ratio of the greedy algorithm for the metric Traveling Salesman Problem is $\Theta(\log n)$. Moreover, we prove that the same result also holds for graphic, Euclidean, and rectilinear instances of the Traveling Salesman Problem. Finally we show that the approximation ratio of the Clarke-Wright savings heuristic for the metric Traveling Salesman Problem is $\Theta(\log n)$.