A Structural Equivalence of Symmetric TSP to a Constrained Group Steiner Tree Problem

TL;DR

This paper establishes a structural equivalence between the symmetric Traveling Salesman Problem and a constrained Group Steiner Tree Problem on a bipartite incidence graph, providing new insights into their relationship.

Contribution

It introduces a novel equivalence between symmetric TSP and a constrained Group Steiner Tree Problem on a simplicial incidence graph.

Findings

Maximizing net weight in cGSTP corresponds to minimizing TSP tour length.

The equivalence is based on a bipartite incidence graph between triangles and edges.

A boundary cycle is induced by selecting admissible, disk-like sets of triangles.

Abstract

We present a brief structural equivalence between the symmetric TSP and a constrained Group Steiner Tree Problem (cGSTP) defined on a simplicial incidence graph. Given the complete weighted graph on the city set V, we form the bipartite incidence graph between triangles and edges. Selecting an admissible, disk-like set of triangles induces a unique boundary cycle. With global connectivity and local regularity constraints, maximizing net weight in the cGSTP is exactly equivalent to minimizing the TSP tour length.