The average solution of a TSP instance in a graph

TL;DR

This paper introduces the concept of average k-TSP distance in graphs, relating it to Steiner distances, and provides bounds and characterizations for these averages based on graph order.

Contribution

It defines the average k-TSP distance, explores its relation to Steiner distances, and establishes bounds and equality conditions for various graph sizes.

Findings

Relation between average k-TSP and Steiner distances

Characterization of equality cases in bounds

Sharp bounds for average k-TSP based on graph order

Abstract

We define the average $k$-TSP distance $\mu_{tsp,k}$ 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 $\mu_{tsp,k}(G)$ given the order of the graph.