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

2024-10-14

International Symposium on Algorithms and Computation

AI Summary

This paper studies the δ-Tour problem on graphs under the continuous edge model: find the shortest closed walk such that every point on every edge lies within distance δ of the walk. This problem unifies the Chinese Postman Problem (δ = 0) and the Graph TSP (δ = 1/2) along a continuous spectrum. Using techniques from combinatorial optimization, parameterized algorithms, and ETH-based reductions, we establish a fine-grained parametric complexity characterization. Our main contribution is the discovery of a three-stage computational phase transition in δ: for δ < 3/2, the problem is APX-hard yet admits a constant-factor approximation; for δ ≥ 3/2, an O(log n)-approximation is achievable and tight; and δ = 3/2 marks a sharp threshold where the problem becomes FPT/W[2]-hard. As a corollary, we obtain a new APX-hardness result for the Graph TSP on cubic bipartite graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Security and Privacy: Large-scale security measurements

Abstract

A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. For $delta geq 0$, we introduce the problem $delta$-Tour, where the objective is to find the shortest tour that comes within a distance of $delta$ 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 $delta$-Tour for other values of $delta$, noting that the problem's behavior and the insights required to understand it differ significantly across various $delta$ regimes. On one hand, we examine the approximability of the problem for every fixed $delta>0$: (1) For every fixed $0<delta<3/2$, the problem $delta$-Tour admits a constant-factor approximation and is APX-hard, while for every fixed $delta geq 3/2$, the problem admits an $O(log n)$-approximation algorithm 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 tour length, it is relatively easy to show that 3/2 is the threshold of fixed-parameter tractability: (2) For every fixed $0<delta<3/2$, the problem $delta$-Tour is FPT parameterized by tour length but is W[2]-hard for every fixed $delta geq 3/2$. On the other hand, if $delta$ is part of the input, then an interesting phenomenon occurs when $delta$ is a constant fraction of n: (3) Here, the problem can be solved in time $f(k) n^{O(k)}$, where $k = lceil n/delta ceil$; however, assuming ETH, there is no algorithm that solves the problem in time $f(k) n^{o(k/log k)}$.

Problem

• Develops polynomial-time approximation algorithms for δ-Tour.
• Explores shortest tours covering all points on edges.
• Analyzes δ-Tour behavior across different δ regimes.

Innovation

• Polynomial-time approximation algorithms
• Constant-factor approximation for δ-Tour
• O(log n)-approximation for δ ≥ 3/2