Continuous relaxations for the traveling salesman problem

Tuhin Sahai*, Adrian Ziessler, Stefan Klus, Michael Dellnitz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 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 on average requires fewer k-opt moves than existing approaches. Broadly, this work connects dynamical systems theory with combinatorial optimization to provide algorithmic and computational complexity insights.

Original languageEnglish
Pages (from-to)2003-2022
Number of pages20
JournalNonlinear Dynamics
Issue number4
Publication statusPublished - Sep 2019


  • Dynamical systems for optimization
  • Procrustes solution
  • Stable manifolds
  • Traveling salesman problem heuristic

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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