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From Chinese Postman to Salesman and Beyond: Shortest Tour δ-Covering All Points on All Edges

Authors Fabian Frei , Ahmed Ghazy , Tim A. Hartmann , Florian Hörsch , Dániel Marx

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Fabian Frei
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Ahmed Ghazy
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarland University, Saarbrücken, Germany
Tim A. Hartmann
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Florian Hörsch
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

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Fabian Frei, Ahmed Ghazy, Tim A. Hartmann, Florian Hörsch, and Dániel Marx. From Chinese Postman to Salesman and Beyond: Shortest Tour δ-Covering All Points on All Edges. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.31

Abstract

A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For δ ≥ 0, we introduce the problem δ-Tour, where the objective is to find the shortest tour that comes within a distance of δ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate δ-Tour for other values of δ, noting that the problem’s behavior and the insights required to understand it differ significantly across various δ regimes. On the one hand, we first examine the approximability of the problem for every fixed δ > 0:
1) For every fixed 0 < δ < 3/2, the problem δ-Tour admits a constant-factor approximation and is APX-hard, while for every fixed δ ≥ 3/2, the problem admits an O(log n)-approximation in polynomial time and has no polynomial-time o(log n)-approximation, unless P = NP.
Our techniques also yield a new APX-hardness result for graphic TSP on cubic bipartite graphs. When parameterizing by the length of a shortest tour, it is relatively easy to show that 3/2 is the threshold of fixed-parameter tractability:
2) For every fixed 0 < δ < 3/2, the problem δ-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed δ ≥ 3/2.
On the other hand, if δ is considered to be part of the input, then an interesting nontrivial phenomenon appears when δ is a constant fraction of the number of vertices:
3) If δ is part of the input, then the problem can be solved in time f(k)n^O(k), where k = ⌈n/δ⌉; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time f(k)n^o(k/log k).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Chinese Postman Problem
  • Traveling Salesman Problem
  • Continuous Graphs
  • Approximation Algorithms
  • Inapproximability
  • Parameterized Complexity

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