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Disjoint Tours and the Price of Diversity

Mark de Berg, Andrés López Martínez, Frits Spieksma

TL;DR

The paper introduces the Price of Diversity (PoD) for disjoint-tour variants of the Traveling Salesman Problem, formalizing $PoD(\Pi_k)$, $PoD^\infty(\Pi_k)$, and related loss-ratio measures to quantify the overhead of enforcing diversity. It analyzes two problems, Path TSP2 and TSP2, across metric spaces, presenting tight PoD bounds and constructive algorithms for the uniform 1D line $R1n$, the circle $S1n$, and general metric spaces. On the uniform 1D line and circle, it proves $PoD=\frac{13}{7}$ with asymptotic $\frac{8}{5}$ for both problems (via algorithms Paths and Tours), while in general metrics the PoD lower bounds are $3-\varepsilon$ for Path TSP2 and $2-\varepsilon$ for TSP2, with matching simple algorithms achieving these ratios. The results are accompanied by lower-bound constructions, algorithmic guarantees, and discussions of efficiency and open questions (e.g., extending to $k>2$).

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.

Disjoint Tours and the Price of Diversity

TL;DR

The paper introduces the Price of Diversity (PoD) for disjoint-tour variants of the Traveling Salesman Problem, formalizing , , and related loss-ratio measures to quantify the overhead of enforcing diversity. It analyzes two problems, Path TSP2 and TSP2, across metric spaces, presenting tight PoD bounds and constructive algorithms for the uniform 1D line , the circle , and general metric spaces. On the uniform 1D line and circle, it proves with asymptotic for both problems (via algorithms Paths and Tours), while in general metrics the PoD lower bounds are for Path TSP2 and for TSP2, with matching simple algorithms achieving these ratios. The results are accompanied by lower-bound constructions, algorithmic guarantees, and discussions of efficiency and open questions (e.g., extending to ).

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.

Paper Structure

This paper contains 17 sections, 18 theorems, 4 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our problems and results.
  3. Related Work.
  4. Preliminaries
  5. The PoD Framework.
  6. Notation.
  7. The Price of Diversity of $Path TSP2$ in $R1n$
  8. A Lower Bound
  9. An Algorithm for $Path TSP2$
  10. Analysis.
  11. The Price of Diversity of $TSP2$ in $S1n$
  12. A Lower Bound
  13. An Algorithm for $TSP2$
  14. The PoD of $Path TSP2$ and $TSP2$ in $M$
  15. Lower bounds
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

$PoD(Path TSP2{}) = \frac{13}{7}$, and $PoD^\infty(\textsc{Disjoint-Path}$$\textsc{TSP}_2) = \frac{8}{5}$ for the class of uniform metric graphs in $\mathbb{R}^1$ when $s = 1$ and $t = n$.

Figures (1)

  • Figure 1: An optimal pair of edge-disjoint Hamiltonian paths with $cost(H_1, H_2) = 16$ for $n = 11$.

Theorems & Definitions (23)

  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • corollary thmcountercorollary
  • ...and 13 more