The bipartite travelling salesman problem: A pyramidally solvable case

TL;DR

This paper investigates a special case of the bipartite travelling salesman problem with Van der Veen distance matrices, showing it is polynomially solvable under specific coloring conditions using pyramidal tours.

Contribution

It identifies a new polynomial-time solvable case of the BTSP restricted to Van der Veen matrices with a specific coloring pattern, despite the general problem being NP-hard.

Findings

The problem remains NP-hard in general.

A polynomial-time solution exists for a specific coloring pattern.

Optimal solutions can be found in O(n^2) time among pyramidal tours.

Abstract

In the bipartite travelling salesman problem (BTSP), we are given $n=2k$ cities along with an $n\times n$ distance matrix and a partition of the cities into $k$ red and $k$ blue cities. The task is to find a shortest tour which alternately visits blue and red cities. We consider the BTSP restricted to the class of so-called Van der Veen distance matrices. We show that this case remains NP-hard in general but becomes solvable in polynomial time when all vertices with odd indices are coloured blue and all with even indices are coloured red. In the latter case an optimal solution can be found in $O(n^2)$ time among the set of pyramidal tours.