Probabilistic Analysis of Edge Elimination for Euclidean TSP

TL;DR

This work shows that after the edge elimination procedure of Hougardy and Schroeder, the expected number of remaining edges is [Formula: see text], whereas after the nonrecursive part of Jonker and Volgenant, the expected number of remaining edges is [Formula: see text].

Abstract

One way to speed up the calculation of optimal TSP tours in practice is eliminating edges that are certainly not in the optimal tour as a preprocessing step. In order to do so several edge elimination approaches have been proposed in the past. In this work we investigate two of them in the scenario where the input consists of $n$ independently distributed random points in the 2-dimensional unit square with bounded density function from above and below by arbitrary positive constants. We show that after the edge elimination procedure of Hougardy and Schroeder the expected number of remaining edges is $Θ(n)$, while after that the non-recursive part of Jonker and Volgenant the expected number of remaining edges is $Θ(n^2)$.

Figures (16)

• Figure 1: For every edge $pq$ and vertex $r\not \in\{p,q\}$ we construct the sets $I_{p}^{qr}$ and $I_{q}^{pr}$. If $pq$ is in the optimal tour, the neighbors of $r$ in the optimal tour have to lie in $I_{p}^{qr} \cup I_{q}^{pr}$.
• Figure 2: If $pq$ and $rt$ are part of a tour, then either replacing the two edges by $pr$ and $qt$ or $pt$ and $rq$ will result in a new tour.
• Figure 3: Given an edge $pq$, a $\delta>0$ and a vertex $x\neq p,q$ the cones $R_p^{qx}(\delta)$ and $R_q^{px}(\delta)$ that are constructed from $I_p^{qx}$ and $I_q^{px}$ and the circle with radius $\delta$ around $x$.
• Figure 4: Condition (4) states that the blue edges are in total shorter than the red edges.
• Figure 5: The four conditions ensure that the blue edges are in total shorter than the red edges.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 2.1: Non-recursive version of Theorem 1 in Jonker
• Definition 2.2: Section 3 in Hougardy
• Theorem 2.3: Main Edge Elimination (for $\delta=\delta_r=\delta_s$), Theorem 3 & Lemma 12 in Hougardy
• Lemma 3.1
• proof
• Definition 3.2