An effective implementation of the Lin–Kernighan traveling salesman heuristic

Models, relaxations and exact approaches for the capacitated vehicle routing problem

Network Optimization: Continuous and Discrete Models

http://web.ist.utl.pt/~ist11038/CD_Casquilho/TSP1992EJOR_Laporte.pdf

The traveling salesman problem: An overview of exact and approximate algorithms

http://alindas.com/University/Upload/Docs/Navid/neural%20network/2.pdf

Chaotic simulated annealing by a neural network model with transient chaos

The traveling salesman problem is one of a class of difficult problems in combinatorial optimization that is representative of a large number of important scientific and engineering problems. A survey is given of recent applications and methods for solving large problems. In addition, an algorithm for the exact solution of the asymmetric traveling salesman problem is presented along with computational results for several classes of problems. The results show that the algorithm performs remarkably well for some classes of problems, determining an optimal solution even for problems with large numbers of cities, yet for other classes, even small problems thwart determination of a provably optimal solution.

https://science.sciencemag.org/content/sci/251/4995/754.full.pdf

Exact solution of large asymmetric traveling salesman problems.

A branch and bound algorithm for capacitated vehicle routing is described. Lower bounds are derived by relaxing the subtour elimination and vehicle capacity constraints to yield a perfect b-matching problem. Bounds are strengthened by using Lagrange multipliers to enforce subtour elimination and capacity constraints. Computational results are reported for problems taken from the literature for sizes up to 61 vertices. With the exception of the 48 vertex instance (att48.vrp), the algorithm solves all problems in TSPLIB having 51 or fewer vertices. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A Matching Based Exact Algorithm for Capacitated Vehicle Routing Problems

: We describe a parallel version of the shortest augmenting path algorithm for the assignment problem. While generating the initial dual solution and partial assignment in parallel does not require substantive changes in the sequential algorithm, using several augmenting paths in parallel does require a new dual variable recalculation method. The parallel algorithm was tested on a 14-processor Butterfly Plus computer, on problems with up to 900 million variables. The speedup obtained increases with problem size. The algorithm was also embedded into a parallel branch and bound procedure for the traveling salesman problem on a directed graph, which was test on the Butterfly Plus on problems involving up to 7,500 cities. To our knowledge, these are the largest assignment problems and traveling salesman problems solved so far.

/pdf/a-parallel-shortest-augmenting-path-algorithm-for-the-5eiqweajm3.pdf

A parallel shortest augmenting path algorithm for the assignment problem

We describe an algorithm for finding a minimum cost perfect b-matching in a weighted undirected graph G=(V,E), with arbitrary edge capacities. The algorithm is based on a staged approach that sequentially applies increasingly more expensive steps until a solution is found. The most significant step involves a primal-dual tree growth procedure that selectively activates and deactivates facet inducing blossom constraints. The algorithm progressively eliminates half-integral values from the primal solution without creating any new fractional values. We introduce the concept of frozen supervertices to address previously unknown complications that arise due to the presence of non-unit capacity edges. We discuss the relationship of the algorithm to previously published cutting plane methods. The algorithm is guaranteed to terminate in O(|V|2|E|) steps, although computational results suggest that much less effort is typically required. Computational results show the algorithm to be effective on problems derived ...

A Staged Primal-Dual Algorithm for Perfect b-Matching with Edge Capacities

We describe an algorithm for finding a minimum cost perfect two-matching in a weighted undirected graph, G = (V, E). The algorithm is based on a staged approach that sequentially applies increasingly more expensive steps until a solution is found. First, the degree constrained LP relaxation is posed as a bipartite two-matching problem and solved using a network flow algorithm. Next, the facet inducing two-matching cutset inequalities are activated selectively within a primal-dual framework that progressively eliminates half-integral values from the primal solution while maintaining dual feasibility. The algorithm is guaranteed to terminate in O(|V|2|E|) steps, although computational results suggest this upper bound to be pessimistic. The algorithm is experimentally shown to be effective on TSPLIB test problem[1] ranging in size from 17 to 11,849 vertices. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Donald L. Miller

Papers

Exact solution of large asymmetric traveling salesman problems.

A Matching Based Exact Algorithm for Capacitated Vehicle Routing Problems

A parallel shortest augmenting path algorithm for the assignment problem

A Staged Primal-Dual Algorithm for Perfect b-Matching with Edge Capacities

A Staged Primal-Dual Algorithm for Finding a Minimum Cost Perfect Two-Matching in an Undirected Graph