The Node Weight Dependent Traveling Salesperson Problem: Approximation Algorithms and Randomized Search Heuristics

TL;DR

This paper introduces a weight-dependent variant of the Traveling Salesperson Problem, providing approximation algorithms and experimental heuristics to understand how node weights influence solution quality and complexity.

Contribution

It presents the first approximation algorithm for the weighted TSP with metric distances and bounded weights, and evaluates randomized heuristics on this problem variant.

Findings

Approximation ratio of 3.59 for the weighted TSP.

Node weights significantly affect tour structure.

Heuristics' performance varies with weight distribution.

Abstract

Several important optimization problems in the area of vehicle routing can be seen as a variant of the classical Traveling Salesperson Problem (TSP). In the area of evolutionary computation, the traveling thief problem (TTP) has gained increasing interest over the last 5 years. In this paper, we investigate the effect of weights on such problems, in the sense that the cost of traveling increases with respect to the weights of nodes already visited during a tour. This provides abstractions of important TSP variants such as the Traveling Thief Problem and time dependent TSP variants, and allows to study precisely the increase in difficulty caused by weight dependence. We provide a 3.59-approximation for this weight dependent version of TSP with metric distances and bounded positive weights. Furthermore, we conduct experimental investigations for simple randomized local search with classical mutation operators and two variants of the state-of-the-art evolutionary algorithm EAX adapted to the weighted TSP. Our results show the impact of the node weights on the position of the nodes in the resulting tour.