This is a preliminary version of a chapter that appeared in the book Local Search in Combinatorial Optimization, E H L Aarts and J K Lenstra (eds), John Wiley and Sons, London, 1997, pp 215-310 The traveling salesman problem (TSP) has been an early proving ground for many approaches to combinatorial optimization, including classical local optimization techniques as well as many of the more recent variants on local optimization, such as simulated annealing, tabu search, neural networks, and genetic algorithms This chapter discusses how these various approaches have been adapted to the TSP and evaluates their relative success in this perhaps atypical domain from both a theoretical and an experimental point of view

The Traveling Salesman Problem: A Case Study in Local Optimization

The Traveling Salesman Problem (TSP) is often cited as the prototypical “hard” combinatorial optimization problem. As such, it would seem to be an ideal candidate for nonstandard algorithmic approaches, such as simulated annealing, and, more recently, genetic algorithms. Both of these approaches can be viewed as variants on the traditional technique called local optimization. This paper surveys the state of the art with respect to the TSP, with emphasis on the performance of traditional local optimization algorithms and their new competitors, and on what insights complexity theory does, or does not, provide.

Local Optimization and the Traveling Salesman Problem

It has been over a decade since neural networks were first applied to solve combinatorial optimization problems. During this period, enthusiasm has been erratic as new approaches are developed and (sometimes years later) their limitations are realized. This article briefly summarizes the work that has been done and presents the current standing of neural networks for combinatorial optimization by considering each of the major classes of combinatorial optimization problems. Areas which have not yet been studied are identified for future research.

/pdf/neural-networks-for-combinatorial-optimization-a-review-of-c2zr5iyu7l.pdf

Neural Networks for Combinatorial Optimization: a Review of More Than a Decade of Research

A simple formulation of the TSP energy function is described which, in combination with a normalized Hopfield-Tank neural network, eliminates the difficulty in finding valid tours. This technique is applicable to many other optimization problems involving n-way decisions (such as VLSI layout and resource allocation) and is easily implemented in a VLSI neural network. The solution quality is shown to be dependent on the formation of seed-points which are influenced by the constraint penalties and the temperature (i.e. the neural gain). Near-optimal tours are found by annealing the network down to a critical temperature at which a single seed-point is dominant. The seed-points and critical temperature (which also affect standard Hopfield network solutions to the TSP) can be predicted with reasonable accuracy. It is also shown that the annealing process is not necessary and good tours result if the network is allowed to converge solely at the critical temperature. The seed-points can be eliminated entirely by assigning different temperatures to groups of neurons such that the tour evolves uniformly throughout the cities. The resulting network finds the optimum tour in a 30-city example in 30% of the trials.

Improving the performance of the Hopfield-Tank neural network through normalization and annealing

Reasoning, nonmonotonicity and learning in connectionist networks that capture propositional knowledge

The authors demonstrate that Hopfield-type networks can find reasonable solutions to the traveling salesman problem (TSP) and the optimal list-matching problem (LMP). They show how to avoid the difficulties encountered by G.V. Wilson and G.S. Pawley (1988) by using a modified energy functional which yields better solutions to the TSP than J.J. Hopfield and D.W. Tank's (1985) original formulation. In addition, two fixed-parameter networks are described, one for the TSP and the other for the LMP. The performance of the network for the TSP is comparable to the performance of the modified energy functional formulation, while the network for the list-matching problem is shown to perform better than a simple heuristic method. A major feature of these two networks is that the problem-dependent cost data are contained entirely in the linear term of the energy functional-the quadratic part contains only constraint information. This feature has the advantage that all costs can be presented to the network as inputs rather than as connection weights, much reducing hardware complexity. >

Alternative networks for solving the traveling salesman problem and the list-matching problem

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Alternative networks for solving the traveling salesman problem and the list-matching problem