# Subtour Elimination Constraints Imply a Matrix-Tree Theorem SDP Constraint for the TSP

## TL;DR

This paper demonstrates that a semidefinite programming constraint for the TSP, based on the matrix-tree theorem, is implied by the subtour elimination constraints, linking SDP and LP relaxations.

## Contribution

It shows that the matrix-tree theorem-based SDP constraint is implied by the subtour elimination linear constraints in TSP relaxations.

## Key findings

- SDP constraint holds for any weighted 2-edge-connected graph.
- The SDP constraint is implied by subtour elimination linear constraints.
- Finite linear inequalities imply the SDP constraint.

## Abstract

De Klerk, Pasechnik, and Sotirov give a semidefinite programming constraint for the Traveling Salesman Problem (TSP) based on the matrix-tree Theorem. This constraint says that the aggregate weight of all spanning trees in a solution to a TSP relaxation is at least that of a cycle graph. In this note, we show that the semidefinite constraint holds for any weighted 2-edge-connected graph and, in particular, is implied by the subtour elimination constraints of the subtour elimination linear program. Hence, this semidefinite constraint is implied by a finite set of linear inequality constraints.