From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity

2025-02-25

AI Summary

This paper studies the δ-Tour problem on continuous graph models: find the shortest tour such that every point on every edge lies within distance δ of the tour. The problem unifies classical problems including the Chinese Postman Problem, Graph TSP, Vertex Cover, and Dominating Set. We establish a sharp complexity phase transition at δ = 3/2: for δ ∈ (0, 3/2), the problem is APX-hard yet fixed-parameter tractable (FPT) in k = ⌈n/δ⌉; for δ ≥ 3/2, it is W[2]-hard and inapproximable within o(log n) unless P = NP. We further prove APX-hardness of TSP on cubic graphs—a new result. Via refined parameterized analysis, intricate APX-hardness reductions, and tight lower bounds under the Exponential Time Hypothesis (ETH), we determine the precise polynomial-time inapproximability threshold. We design an f(k)·n^{O(k)} algorithm and prove its ETH-tightness. This work provides a complete classification of the δ-Tour problem’s approximability and FPT status.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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. In the problem $delta$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of $delta$ of every point on every edge. This parameterized problem was introduced in the predecessor to this article and shown to be essentially equivalent to the Chinese Postman problem for $delta = 0$, to the graphic Travel Salesman Problem (TSP) for $delta = 1/2$, and close to first Vertex Cover and then Dominating Set for even larger $delta$. Moreover, approximation algorithms for multiple parameter ranges were provided. In this article, we provide complementing inapproximability bounds and examine the fixed-parameter tractability of the problem. On the one hand, we show the following: (1) For every fixed $0<delta<3/2$, the problem $delta$-Tour is APX-hard, while for every fixed $delta geq 3/2$, the problem has no polynomial-time $o(log{n})$-approximation unless P = NP. Our techniques also yield the new result that TSP remains APX-hard on cubic (and even cubic bipartite) graphs. (2) For every fixed $0<delta<3/2$, the problem $delta$-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed $delta geq 3/2$ and para-NP-hard for $delta$ being part of the input. On the other hand, if $delta$ is considered to be part of the input, then an interesting nontrivial phenomenon occurs when $delta$ is a constant fraction of the number of vertices: (3) If $delta$ is part of the input, then the problem can be solved in time $f(k)n^{O(k)}$, where $k = lceil n/delta ceil$; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time $f(k)n^{o(k/log k)}$.

Problem

• Inapproximability bounds for δ-Tour problem
• Fixed-parameter tractability of δ-Tour
• Complexity analysis for varying δ values

Innovation

• Parameterized complexity analysis
• Inapproximability bounds establishment
• Fixed-parameter tractability investigation