# Continuous Relaxations for the Traveling Salesman Problem

## TL;DR

This paper introduces a novel approach to the TSP by embedding it into manifolds using dynamical systems, providing a biasing method for heuristics that often reduces the number of local search steps needed.

## Contribution

It develops a new relaxation technique for the TSP on the manifold of orthogonal matrices and integrates it with the Lin--Kernighan heuristic to improve solution biasing.

## Key findings

- Procrustes-based solutions often converge to undesirable equilibria.
- The approach biases the Lin--Kernighan heuristic towards better solutions.
- Fewer k-opt moves are needed compared to traditional methods.

## Abstract

In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the relaxed discrete optimization problem into appropriate manifolds. We explore multiple embedding techniques -- namely, the construction of new dynamical systems on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. Using these dynamical systems, we analyze the local neighborhood around the optimal TSP solutions (which are equilibria) using computations to approximate the associated \emph{stable manifolds}. We find that these flows frequently converge to undesirable equilibria. However, the solutions of the dynamical systems and the associated Procrustes approximation provide an interesting biasing approach for the popular Lin--Kernighan heuristic which yields fast convergence. The Lin--Kernighan heuristic is typically based on the computation of edges that have a `high probability' of being in the shortest tour, thereby effectively pruning the search space. Our new approach, instead, relies on a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the Lin--Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than existing methods, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes based approach on several TSP instances and find that our approach often requires fewer $k$-opt moves than existing approaches. Broadly, we hope that this work initiates more work in the intersection of dynamical systems theory and combinatorial optimization.