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

TL;DR

This work analyzes δ-Tour in the continuous graph model, establishing a sharp complexity transition at $δ=\tfrac{3}{2}$ and presenting a tight blend of inapproximability and parameterized results. It shows APX-hardness for every fixed $δ∈(0,\tfrac{3}{2})$ and log-approximation barriers for $δ≥\tfrac{3}{2}$, with TSP APX-hardness on cubic bipartite graphs as a corollary; it also provides both FPT algorithms (for $δ<\tfrac{3}{2}$) parameterized by tour length and hardness results (W[2]-hard, para-NP-hard) for larger $δ$, plus an ETH-based hardness dominance when $δ$ is allowed to vary. The paper further studies the regime where $δ$ is a part of the input, giving an $f(k)n^{O(k)}$-time algorithm for $k=⌈n/δ⌉$ and proving ETH-based lower bounds, along with XP algorithms in $n/δ$ and W[1]-hardness with respect to the same parameter. Central to the results are discretization and “nice tour” structural results, which enable reductions from Vertex-Cover, Dominating Set, and Binary-CSP and connect δ-Tour to classical graph problems. Overall, the findings illuminate the thresholds and computational boundaries for covering continuous graphs with minimal δ-tours, guiding algorithm design under varying δ regimes.

Abstract

A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. In the problem $δ$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of $δ$ 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 $δ= 0$, to the graphic Travel Salesman Problem (TSP) for $δ= 1/2$, and close to first Vertex Cover and then Dominating Set for even larger $δ$. 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 $0 < δ< 3/2$, the problem $δ$-Tour is APX-hard, while for every fixed $δ\geq 3/2$, 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 $0 < δ< 3/2$, 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 $δ\geq 3/2$ and para-NP-hard for $δ$ being part of the input. On the other hand, if $δ$ is considered to be part of the input, then an interesting nontrivial phenomenon occurs 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 $k = \lceil n/δ\rceil$; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time $f(k)n^{o(k/\log k)}$.

Figures (5)

• Figure 1: The four ways a nice ${\delta\textup{-tour}}$ defined by at least $3$ points can interact with an edge $uv$.
• Figure 2: An $(\alpha, \beta, k)$-chain gadget $G$.
• Figure 5: Constraint gadget $\Gamma_{e}$ for $e = uv \in E(G)$ as defined in the proof of \ref{['thm:csp_hardness']}.
• Figure 6: An example of the connection between gadgets $\Gamma_{e_1}$, $\Gamma_{e_2}$, and $\Gamma_{e_3}$ sharing a variable $v$. Oscillating lines are paths of length $\delta-4n$, and bold lines are paths of length $2n$. The vertices within gadgets indicate candidate vertices.
• Figure 7: The graph $G'$. The figure shows an example of how three gadgets sharing a variable connect. The remaining connectors and middle paths are omitted for clarity. Bold oscillating lines are paths of length $\delta-4n+1$, and bold lines are paths of length $2n$.

Theorems & Definitions (45)

• Theorem 1: Approximation Algorithms FreiGHHM24
• Theorem 2: APX-Hardness
• Theorem 3
• Theorem 4: Log-APX-Hardness
• Theorem 5: Natural Parameterization
• Theorem 6: Classic Complexity for Fixed Tour Length
• Lemma 7: Nice Tours FreiGHHM24
• Lemma 8: Discretization Lemma FreiGHHM24
• Proposition 9
• Lemma 10: Stronger Discretization Lemma FreiGHHM24