Given a set S of n points in the unit square $[ 0,1 ]^d $, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the traveling salesman problem in the unit square is a point set $S^{( n )} $ whose optimal traveling salesman tour achieves the maximum possible length among all point sets $S \subset [ 0,1 ]^d $, where $| S | = n$. An open prob...

We relate the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope to a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron. This relationship is quite surprising. The proof is elegant and geometric: it makes use of recent results on “flattening” parts of the boundary complex of the polar of the Graphical Traveling Salesman Polyhe...

In this paper, we present a polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). The proposed linear program is a network flow-based model. Numerical implementation issues and results are discussed. KeyWords: Linear Programming; Network Optimization; Integer Programming; Traveling Salesman Problem; Combinatorial Optimization; Scheduling; Sequencing.

In this paper, we present a polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). The proposed linear program is a network flow-based model. Numerical implementation issues and results are discussed. Key-Words: Linear Programming; Network Optimization; Integer Programming; Traveling Salesman Problem; Combinatorial Optimization; Scheduling; Sequencing. (Forthco...

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The k-traveling salesman problem, or k-TSP is: given a graph with edge weights and an integer k, find a simple cycle of minimum weight visiting exactly k nodes. To obtain lower bounds for the traveling salesman problem the 2-matching relaxation and the 1-tree relaxation can be used. We generalize these two relaxations for the k-TSP.

Abstract In this short note, the dual problem for the traveling salesman problem is constructed through the classic Lagrangian. The existence of optimality conditions is expressed as a corresponding inverse problem. A general 4-cities instance is given, and the numerical experiment shows that the classic Lagrangian may not be applicable to the traveling salesman problem.

By introducing the concept of Oracle we propose an approach for improving the performance of genetic algorithms for large-scale asymmetric Traveling Salesman Problems. The results have shown that the proposed approach allows overcoming some traditional problems for creating efficient genetic algorithms. Keywords—Genetic algorithms, Traveling Salesman Problem, optimal decision distribution, Oracle.

This note aims to give an application of entropy to the analysis of algorithms, using Shearer’s entropy lemma to bound the running time of Bellman’s dynamic programming solution to the traveling salesman problem. The original venue for this material was my course Randommised algorithms (2008) at Lund University. 1. The Traveling Salesman Problem Given a weighted graph like