Counting Locally Optimal Tours in the TSP

TL;DR

It is conjecture that the true bound of the expected number of 2-optimal tours in random instances of the TSP on complete graphs is at most $O(1.2098^n \sqrt{n!})$.

Abstract

We show that the problem of counting the number of 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is $O(1.2098^n \sqrt{n!})$. Based on numerical experiments, we conjecture that the true bound is at most $O(\sqrt{n!})$, which is approximately the square root of the total number of tours.

Figures (3)

• Figure 1: Schematic depiction of the reduction we use to prove #P-hardness of #2Opt, and of a 2-optimal tour in the image instance.
• Figure 2: Colors of the edges in the tour $T$ at stage $t = 3$, for $n = 2^4 + 1$. The dotted lines are drawn to show the boundaries of each segment $T_i$ more clearly. The segments of the tour are numbered starting at the right and proceeding counterclockwise. The 2-changes we consider in the proof of \ref{['lemma:set chord disjoint']} are then the 2-changes formed from the red edges in $T_i$ (drawn black) and the blue edges in $T_{i+1}$ (drawn gray) that appear in even positions along $T$, for $i$ odd. Note that the last edge along $T$ is drawn dashed to indicate that it is not used in the construction of $\mathcal{S}$.
• Figure 3: Estimated volume of the 2-opt polytope $P_n$, for different values of $n$, as computed by Volesti. For comparison, the function $n \mapsto 1/\sqrt{n!}$ is also plotted.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1
• Definition 2
• Theorem 2.1: amemiyaMultivariateRegressionSimultaneous1974
• Theorem 2.2: b.g.MomentsCalculationDouble2009
• Lemma 3.1
• proof