The frequency $K_i$s for symmetrical traveling salesman problem

TL;DR

The paper develops a frequency Ki framework to identify optimal Hamiltonian cycle (OHC) edges in symmetric TSP by extending prior frequency analysis from K4 to Ki for i ∈ [4,n]. It defines optimal i-vertex paths OP^i within each K_i, builds Ki by counting edge containment in OP^i, and proves strong separation between OHC and ordinary edges via bounds such as $f_i(OHC)> \frac{1}{2}{i\choose 2}$ and $f_i(ord)<\frac{1}{2}{i\choose 2}$ with average frequencies $E[f(OHC)]>\frac{i^2-4i+7}{2}$ and $E[f(ord)]<2$, respectively, with peak frequency at $P_0$ where $P_0=\frac{n}{2}+2$ (even) or $P_0=\frac{n+1}{2}+1$ (odd). Extending to Kn by aggregating across Ki yields lower bounds on edge frequencies and a dynamic programming algorithm that finds all OHC edges in time $O(n^2 2^{0.3660n})$, enabling efficient OHC discovery for large TSP instances. The authors validate the theory with extensive experiments on small and real-world TSP instances, demonstrating pronounced frequency/probability gaps between OHC and ordinary edges and showcasing the practical utility of Ki-based filtering for speeding up OHC identification. The results highlight that exploiting combinatorial path-frequency structure can complement distance-based heuristics in TSP edge classification and exact recovery tasks.

Abstract

The frequency $K_i$s ($i\in[4,n]$) are studied for symmetrical traveling salesman problem ($TSP$) to identify the edges in optimal Hamiltonian cycle ($OHC$). A frequency $K_i$ is computed with a sort of ${{i}\choose{2}}$ optimal $i$-vertex paths with given endpoints (optimal $i$-vertex path) in a corresponding $K_i$ in $K_n$. In frequency $K_i$, the frequency of an edge is the number of the optimal $i$-vertex paths containing the edge in the corresponding $K_i$. Given an $OHC$ edge related to $K_i$, it has a frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the corresponding frequency $K_i$, and that of an ordinary edge not in $OHC$ is smaller than $\frac{1}{2}{{i}\choose{2}}$. On average, an $OHC$ edge in $K_i$ has the expected frequency bigger than $\frac{i^2-4i+7}{2}$ whereas an ordinary edge has the expected frequency smaller than 2. Moreover, given a frequency $K_i$ containing an $OHC$ edge related to $K_n$, the frequency of the $OHC$ edge is bigger than $\frac{1}{2}{{i}\choose{2}}$ in the worst average case. It implies that the average frequency of an $OHC$ edge computed with frequency $K_i$s is bigger than $\frac{1}{2}{{i}\choose{2}}$. It also found that the probability that an $OHC$ edge is contained in optimal $i$-vertex paths keeps stable or increases according to $i\in [4, n]$. As the frequency $K_i$s are used to compute the frequency of an edge, each $OHC$ edge has its own peak frequency at $i=P_0$ where $P_0=\frac{n}{2} + 2$ for even $n$ or $\frac{n+1}{2} + 1$ for odd $n$. For ordinary edges out of $OHC$, the probability that they are contained in optimal $i$-vertex paths decreases according to $i$. Moreover, the average frequency of an ordinary edge will be smaller than $\frac{1}{2}{{i}\choose{2}}$ if $i \geq [0.3660n + 5.5849]$. Based on these findings, an algorithm is presented to find $OHC$ in $O(n^62^{0.3660n})$ time using dynamic programming.

Figures (19)

• Figure 1: The illustration of $OHC$ in $K_n$.
• Figure 2: The ${{i}\choose{2}}$$OP^i$s in one $K_i$.
• Figure 3: The two $G_5$s contained in the $G_i$ containing all edges in the ${{i}\choose{2}}$$OP^i$s in $K_i$.
• Figure 4: The changes of $r$ and $1-r$ according to $\epsilon \in [2\times 10^{-6}, 1]$ for $n=1000$.
• Figure 5: The percents of $J$, $K$, $L$ and $J+L$ according to $i\in [4, 1000]$ for $n=1000$.

Theorems & Definitions (18)

• theorem 3.1
• proof
• theorem 3.2
• proof
• theorem 3.3
• proof
• theorem 3.4
• proof
• theorem 3.5
• proof