# A Note on the Polytope of Bipartite TSP

## TL;DR

This paper investigates the structure of the bipartite TSP polytope, revealing that many comb inequalities are not facet defining in the bipartite case, unlike in the general TSP, due to specific vertex class conditions.

## Contribution

It characterizes the differences in the polytope structure of bipartite TSP, especially regarding the facet-defining properties of comb inequalities.

## Key findings

- Many comb inequalities are satisfied under degree and subtour constraints in bipartite TSP.
- Certain comb inequalities are violated when vertex classes exceed half the intersections.
- The structure of the bipartite TSP polytope differs significantly from the general TSP polytope.

## Abstract

The main result of this paper is that the polytope of the bipartite TSP is significantly different from that of the general TSP. Comb inequalities are known as facet defining ones in the general case. In the bipartite case, however, many of them are satisfied whenever all degree and subtour elimination constraints are satisfied, {\em i.e.}\ these comb inequalities are not facet defining. The inequalities in question belong to the cases where vertices of one of the two classes occur in less than the half of the intersections of the teeth and the hand. Such side conditions are necessary, as simple example shows that the comb inequality can be violated when each class has vertices in more than the half of the intersections.