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Improving polynomial bounds for the Graphical Traveling Salesman Problem with release dates on paths

Thailsson Clementino, Rosiane de Freitas

TL;DR

This work addresses GTSP-rd restricted to path graphs, tackling two objectives: makespan minimization (time) and total travel distance (distance). It develops refined dynamic-programming frameworks that exploit path structure and non-interlacing routes, supported by data-structures such as minqueues and Thorup-style heaps. The main contributions are: $O(n)$ time for GTSP-rd(time) when the depot sits at a path extremity, $O(n \\log \\log n)$ for GTSP-rd(distance) in the same extremity case, and $O(n^2)$ (time) plus $O(n^2 \\log \\log n)$ (distance) for the general depot location on a path. These results yield exact, scalable polynomial-time algorithms for GTSP-rd on paths and suggest avenues for extending these techniques to more complex sparse graphs.

Abstract

The Graphical Traveling Salesman Problem with release dates (GTSP-rd) is a variation of the TSP-rd where each vertex in a weighted graph $G$ must be visited at least once, respecting the release date restriction. The edges may be traversed multiple times if necessary, as in some sparse graphs. This paper focuses on solving the GTSP-rd in paths. We consider two objective functions: minimizing the route completion time (GTSP-rd (time)) and minimizing the total distance traveled (GTSP-rd (distance)). We present improvements to existing dynamic programming algorithms, offering an $O(n)$ solution for paths where the depot is located at the extremity and an $O(n^2)$ solution for paths where the depot is located anywhere. For the GTSP-rd (distance), we propose an $O(n \log \log n)$ solution for the case with the depot at the extremity and an $O(n^2 \log \log n)$ solution for the general case.

Improving polynomial bounds for the Graphical Traveling Salesman Problem with release dates on paths

TL;DR

This work addresses GTSP-rd restricted to path graphs, tackling two objectives: makespan minimization (time) and total travel distance (distance). It develops refined dynamic-programming frameworks that exploit path structure and non-interlacing routes, supported by data-structures such as minqueues and Thorup-style heaps. The main contributions are: time for GTSP-rd(time) when the depot sits at a path extremity, for GTSP-rd(distance) in the same extremity case, and (time) plus (distance) for the general depot location on a path. These results yield exact, scalable polynomial-time algorithms for GTSP-rd on paths and suggest avenues for extending these techniques to more complex sparse graphs.

Abstract

The Graphical Traveling Salesman Problem with release dates (GTSP-rd) is a variation of the TSP-rd where each vertex in a weighted graph must be visited at least once, respecting the release date restriction. The edges may be traversed multiple times if necessary, as in some sparse graphs. This paper focuses on solving the GTSP-rd in paths. We consider two objective functions: minimizing the route completion time (GTSP-rd (time)) and minimizing the total distance traveled (GTSP-rd (distance)). We present improvements to existing dynamic programming algorithms, offering an solution for paths where the depot is located at the extremity and an solution for paths where the depot is located anywhere. For the GTSP-rd (distance), we propose an solution for the case with the depot at the extremity and an solution for the general case.

Paper Structure

This paper contains 12 sections, 13 theorems, 12 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Graphical Traveling Salesman Problem with release dates
  3. GTSP-rd on paths
  4. Special case: depot on an extremity
  5. GTSP-rd (time)
  6. Proposed Solution for GTSP-rd (time) in $O(n)$
  7. GTSP-rd (distance)
  8. Proposed Solution for GTSP-rd (distance) in $O(n \log \log n)$
  9. General case
  10. Solution $O(n^2)$ for GTSP-rd (time)
  11. Solution $O(n^2 \log \log n)$ for GTSP-rd (distance)
  12. Concluding remarks

Key Result

Proposition 1

archetti2015complexity Given two vertices $i$ and $j$ such that $i<j$, if $\tau_i < \tau_j$ then there is exist an optimal solution such that $i$ and $j$ are delivered in the same route.

Figures (3)

  • Figure 1: A GTSP-rd solution, containing the three routes $\mathcal{R}_1,\mathcal{R}_2,\mathcal{R}_3$.
  • Figure 2: An arbitrary Path instance with the depot in an arbitrary vertex.
  • Figure 3: Example of minqueue usage to calculate $L(i,j)$ to some $i \in N_l$.

Theorems & Definitions (15)

  • Proposition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 5 more