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Better approximation guarantee for Asymmetric TSP

Jens Vygen

Abstract

We improve the approximation ratio for the Asymmetric TSP to less than 15. We also obtain improved ratios for the special case of unweighted digraphs and the generalization where we ask for a minimum-cost tour with given (distinct) endpoints. Moreover, we prove better upper bounds on the integrality ratios of the natural LP relaxations.

Better approximation guarantee for Asymmetric TSP

Abstract

We improve the approximation ratio for the Asymmetric TSP to less than 15. We also obtain improved ratios for the special case of unweighted digraphs and the generalization where we ask for a minimum-cost tour with given (distinct) endpoints. Moreover, we prove better upper bounds on the integrality ratios of the natural LP relaxations.
Paper Structure (31 sections, 11 theorems, 40 equations, 2 figures)

This paper contains 31 sections, 11 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Revisiting Vertebrate Pairs and Svensson's Algorithm
  3. Reduction to Vertebrate Pairs
  4. Strongly laminar instances.
  5. Backbone.
  6. ATSP via vertebrate pairs.
  7. Asymmetric Graph TSP.
  8. Main result.
  9. High-Level Description of Svensson's Algorithm
  10. Details of Svensson's Algorithm
  11. Local.
  12. Vertex budgets.
  13. Slack.
  14. Initialization.
  15. Number of restarts.
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\alpha$ be a constant with $\alpha > 3 + 2\sqrt{2}$. Then there is a polynomial-time algorithm that computes, for any given strongly connected digraph $G=(V,E)$ with edge costs $c:E\to\mathbb{R}_{\ge 0}$, a tour with cost at most $(9+\alpha)\cdot \mathrm{LP}$, where $\mathrm{LP}$ denotes the va

Figures (2)

  • Figure 1: Assume for simplicity that $y_v$ is the same for all $v\in V$ and $x(\delta^-(v))=1$ for all $v\in \tilde{W}_i$. The subtour cover can be decomposed into cycles. Among these cycles, consider those that intersect the set $\tilde{W}_i$ but no connected component of the initialization with smaller index. Four of these are sketched in the figure in different colors. If for any of these, the outside part is large compared to the inside part, then one gets a better initialization. In the simple setting, every vertex of $\tilde{W}_i$ belongs to at most two of these cycles, so the total number of vertices (and hence edges) of the cycles is not much more than the total number of vertices in $\tilde{W}_i$.
  • Figure 2: Proof of Lemma \ref{['lemma:Fedges']}: An example of $F^{t_2}_i$ is shown in red. Every circuit $C\in\mathcal{F}^{t_2}_i$ contains an edge of $\delta(V(Z^{t_1}))$.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2: TraV22
  • Theorem 3
  • Theorem 4: TraV25
  • Lemma 5
  • Lemma 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 6 more