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Parameterized Approximation Algorithms for TSP on Non-Metric Graphs

Jingyang Zhao, Zimo Sheng, Mingyu Xiao

TL;DR

This work studies the Traveling Salesman Problem on non-metric graphs through two distance-from-metric parameters, p (violating-triangle vertices) and q (minimum removing set to reach metric). It develops a sequence of fixed-parameter tractable approximation schemes: for p, a simple $( ext{α}+1)$-approximation in $2^{O(p)}$ time, a refined $1.5$-approximation in $2^{O(p\log p)}$ time, and an $( ext{α}+ ext{ε})$-approximation in $n^{O(p/ ext{ε})}$; for q, a $3$-approximation in $2^{O(q\log q)}$ time and an $( ext{α}+ ext{ε})$-approximation in $n^{O(q/ ext{ε})}$ time. The methods rely on carefully partitioning vertices into bad/good sets, constructing constrained spanning trees, and using shortcutting along with new subroutines LIMB, CONNECT, and SHORTCUT to ensure connectivity and parity while controlling weight. Together, these results significantly improve the state of the art for near-metric TSP and open directions for extensions to related problems and parameters.

Abstract

The Traveling Salesman Problem (TSP) is a classic and extensively studied problem with numerous real-world applications in artificial intelligence and operations research. It is well-known that TSP admits a constant approximation ratio on metric graphs but becomes NP-hard to approximate within any computable function $f(n)$ on general graphs. This disparity highlights a significant gap between the results on metric graphs and general graphs. Recent research has introduced some parameters to measure the ``distance'' of general graphs from being metric and explored Fixed-Parameter Tractable (FPT) approximation algorithms parameterized by these parameters. Two commonly studied parameters are $p$, the number of vertices in triangles violating the triangle inequality, and $q$, the minimum number of vertices whose removal results in a metric graph. In this paper, we present improved FPT approximation algorithms with respect to these two parameters. For $p$, we propose an FPT algorithm with a 1.5-approximation ratio, improving upon the previous ratio of 2.5. For $q$, we significantly enhance the approximation ratio from 11 to 3, advancing the state of the art in both cases. In addition, when $p$ (or $q$) is a constant, we obtain a better approximation ratio.

Parameterized Approximation Algorithms for TSP on Non-Metric Graphs

TL;DR

This work studies the Traveling Salesman Problem on non-metric graphs through two distance-from-metric parameters, p (violating-triangle vertices) and q (minimum removing set to reach metric). It develops a sequence of fixed-parameter tractable approximation schemes: for p, a simple -approximation in time, a refined -approximation in time, and an -approximation in ; for q, a -approximation in time and an -approximation in time. The methods rely on carefully partitioning vertices into bad/good sets, constructing constrained spanning trees, and using shortcutting along with new subroutines LIMB, CONNECT, and SHORTCUT to ensure connectivity and parity while controlling weight. Together, these results significantly improve the state of the art for near-metric TSP and open directions for extensions to related problems and parameters.

Abstract

The Traveling Salesman Problem (TSP) is a classic and extensively studied problem with numerous real-world applications in artificial intelligence and operations research. It is well-known that TSP admits a constant approximation ratio on metric graphs but becomes NP-hard to approximate within any computable function on general graphs. This disparity highlights a significant gap between the results on metric graphs and general graphs. Recent research has introduced some parameters to measure the ``distance'' of general graphs from being metric and explored Fixed-Parameter Tractable (FPT) approximation algorithms parameterized by these parameters. Two commonly studied parameters are , the number of vertices in triangles violating the triangle inequality, and , the minimum number of vertices whose removal results in a metric graph. In this paper, we present improved FPT approximation algorithms with respect to these two parameters. For , we propose an FPT algorithm with a 1.5-approximation ratio, improving upon the previous ratio of 2.5. For , we significantly enhance the approximation ratio from 11 to 3, advancing the state of the art in both cases. In addition, when (or ) is a constant, we obtain a better approximation ratio.

Paper Structure

This paper contains 16 sections, 12 theorems, 23 equations, 4 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminary
  4. TSP Parameterized by $p$
  5. The First Algorithm
  6. The Second Algorithm
  7. The Third Algorithm
  8. TSP Parameterized by $q$
  9. The Algorithm
  10. The sub-algorithm: LIMB
  11. The sub-algorithm: CONNECT
  12. The sub-algorithm: SHORTCUT
  13. Conclusion
  14. Two Assumptions
  15. TSP Parameterized by $p$
  16. ...and 1 more sections

Key Result

Lemma 1

It holds that $\hbox{OPT}\geq w(T^*_b)$ and $\hbox{OPT}\geq w(T^*_g)$, where $T^*_b$ (resp., $T^*_g$) denotes an optimal TSP tour in $G[V_b\cup\{o\}]$ (resp., $G[V_g]$).

Figures (4)

  • Figure 1: An illustration of our ALG.2, where each black (resp., white) node denotes a bad (resp., good) vertex. In (a), the edges in $E(F)$ (resp., $E(\mathcal{A})$) are shown in red (resp., black); In (b), the edges in $\mathcal{M}_\mathcal{A}$ are shown in blue; In (c), $\mathcal{T}'_\mathcal{A}$ is obtained by shortcutting repeated edges on the paths in $\mathcal{A}\cap\mathcal{A}'$ (see Lemma \ref{['LB2.3']}), with the new edges shown in purple; In (d), the TSP tour $T_2$ is obtained by taking shortcuts on $\mathcal{T}'_\mathcal{A}$ (see Lemma \ref{['LB2.3']}), with the new edges also shown in purple.
  • Figure 2: An illustration of our notations: the cycle denotes $T^*$; the black, white, and green nodes denote bad vertices, internal vertices, and anchors, respectively. We have $\mathcal{A}=\{x_1...x_4,x_5x_6,x_7\}$, $\mathcal{B}=\{x_4y_1,...,y_8x_1\}$, and $\mathcal{R}=\{y_1...y_4,y_5,y_6y_7y_8\}$. Note that $x_7\in\mathcal{A}$ is a 1-path.
  • Figure 3: An illustration of our ALG.4. Black (resp., white) nodes denote bad (resp., internal) vertices; green nodes are anchors. Each ellipse denotes a tree $F\in\mathcal{F}$, with the upper part indicating its potential set $V_F$. In (a), the edges in $E(\mathcal{A})$ (resp., limbs in $\mathcal{B}$) are shown in black (resp., red); In (b), the guessed limbs in $\mathcal{B}'$ are shown in red, and the white and green nodes incident to red edges denote guessed anchors; In (c), the edges in $E(\mathcal{R})$ are shown in blue; In (d), the edges in $E(\mathcal{R}^*)$ are shown in blue; In (e), the guessed edges in $\mathcal{R}'$ are shown in blue; In (f), the edges in $\mathcal{M}$ are shown in purple.
  • Figure 4: An illustration of the two cases. Each black node denotes a bad vertex, each green or white node denotes a good vertex, each blue node denotes a marked bad vertex; the edges in $E(F)$ are shown in black, the edges in $\mathcal{M}_F$ are shown in purple, and the limbs are shown in red. In (a), the black and purple edges are deleted; In (b), $x'$ is a copy of $x$.

Theorems & Definitions (29)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 19 more