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The average solution of a TSP instance in a graph

Stijn Cambie

TL;DR

The paper introduces the average $k$-TSP distance $\mu_{tsp,k}(G)$ and the $k$-TSP Wiener index $W_{tsp,k}(G)$, linking them to the $k$-Steiner distance and Steiner Wiener index and extending the framework to weighted graphs and digraphs. A central result is the sharp bound $W_{tsp,k}(G) \le 2W_k(G)$ with precise equality criteria, including special cases for $k=2$ and $k=3$ and a tree condition for $k\ge 4$, along with a complementary bound $W_{tsp,k}(G) \le k\binom{n}{k}\mu(G)$ with its own equality cases. The authors also establish the exact identity $W_{tsp,3}(G)=(n-2)W_3(G)$ for all $G$ and derive tight extremal bounds for $\mu_{tsp,k}(G)$, characterizing graphs achieving equality. They extend the discussion to digraphs, relate these results to known Wiener-type indices, and comment on conjectures such as a DeLaViña–Waller analogue, noting its failure for larger $k$. Overall, the work provides a foundational analytic framework for average TSP distances and their connections to Steiner-type indices, with implications for graph design and analysis of TSP instances.

Abstract

We define the average $k$-TSP distance $μ_{tsp,k}$ of a graph $G$ as the average length of a shortest walk visiting $k$ vertices, i.e. the expected length of the solution for a random TSP instance with $k$ uniformly random chosen vertices. We prove relations with the average $k$-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for $μ_{tsp,k}(G)$ given the order of the graph.

The average solution of a TSP instance in a graph

TL;DR

The paper introduces the average -TSP distance and the -TSP Wiener index , linking them to the -Steiner distance and Steiner Wiener index and extending the framework to weighted graphs and digraphs. A central result is the sharp bound with precise equality criteria, including special cases for and and a tree condition for , along with a complementary bound with its own equality cases. The authors also establish the exact identity for all and derive tight extremal bounds for , characterizing graphs achieving equality. They extend the discussion to digraphs, relate these results to known Wiener-type indices, and comment on conjectures such as a DeLaViña–Waller analogue, noting its failure for larger . Overall, the work provides a foundational analytic framework for average TSP distances and their connections to Steiner-type indices, with implications for graph design and analysis of TSP instances.

Abstract

We define the average -TSP distance of a graph as the average length of a shortest walk visiting vertices, i.e. the expected length of the solution for a random TSP instance with uniformly random chosen vertices. We prove relations with the average -Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for given the order of the graph.
Paper Structure (5 sections, 10 theorems, 16 equations, 3 figures)

This paper contains 5 sections, 10 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Relations between the Steiner Wiener and TSP Wiener index
  4. Bounds on the average TSP distance
  5. Conclusion and further directions

Key Result

Theorem 2

For a graph $G$ and integer $k\ge 2$, we have $\mathop{\mathrm{W_{tsp,k}}}\nolimits(G)\le 2\mathop{\mathrm{W}}\nolimits_k(G).$ Equality holds if and only if $k=2$, $k=3$ and $G$ contains no vertices $u,v,w$ for which $2\max\{d(u,v),d(u,w),d(v,w)\}<d(u,v)+d(u,w)+d(v,w)$ and every choice of $3$ shorte

Figures (3)

  • Figure 1: Sketch of shortest paths between the three vertices $(u,v,w)$
  • Figure 2: The digraph $DP_{n,d}$
  • Figure 3: A symmetric tree with order $2d+1$ and diameter $d$ with large $\mathop{\mathrm{\mu_{tsp,k}}}\nolimits(T)$

Theorems & Definitions (23)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • ...and 13 more