Disjoint Tours and the Price of Diversity

TL;DR

This paper explores a variant of the Traveling Salesman Problem involving finding two edge-disjoint tours and analyzes the Price of Diversity, providing bounds in specific scenarios and introducing a framework for future research.

Contribution

It introduces the Price of Diversity framework for disjoint tours in TSP and establishes bounds for this ratio in specific metric spaces.

Findings

Established tight bounds for the Price of Diversity in 1-dimensional scenarios.

Derived bounds for general metric spaces.

Proposed the Price of Diversity as a new analytical framework.

Abstract

We study a variant of the Traveling Salesman Problem, where instead of finding a single tour, we want to find a pair of two edge-disjoint tours whose longer tour is as short as possible. We investigate the Price of Diversity (PoD) for this problem, which is the ratio of the cost of the longer of the two tours and the cost of a single optimal tour, in the worst case over all possible instances. We prove (almost) tight bounds on this quantity for a special 1-dimensional scenario and for general metric spaces. We believe that the Price-of-Diversity framework that we introduce is interesting in its own right, and may lead to follow-up work on other problems as well.