Lower bounds for the symmetric travelling salesman problem from Lagrangean relaxations

A comparison of lower bounds for the symmetric circulant traveling salesman problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on...

Semidefinite Relaxations for the Traveling Salesman Problem

Continuous Relaxations for the Traveling Salesman Problem

In this work, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the discrete optimization problem into continuous spaces. We explore multiple embedding techniques – namely, the construction of dynamical flows on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. In particular, we find that the Procru...

Ant colonies for the travelling salesman problem.

We describe an artificial ant colony capable of solving the travelling salesman problem (TSP). Ants of the artificial colony are able to generate successively shorter feasible tours by using information accumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer simulations demonstrate that the artificial ant colony is capable of generating good solutions to b...

On Random Symmetric Travelling Salesman Problems

Let the edges of the complete graph Kn be assigned independent uniform [0; 1] random edge weights. Let ZTSP and Z2FAC be the weights of the minimum length travelling salesman tour and minimum weight 2-factor respectively. We show that whp jZTSP Z2FAC j = o(1). The proof is via by the analysis of a polynomial time algorithm that nds a tour only a little longer than Z2FAC .