Traveling Salesman Problem Instances Research Articles

Microscopic computation in human brain evolution.

When human psychological performance is viewed in terms of cognitive modules, our species displays remarkable differences in computational power. Algorithmically simple computations are generally difficult to perform, whereas optimal routing or "Traveling Salesman" Problems (TSP) of far greater complexity are solved on an everyday basis. It is argued that even "simple" instances of TSP are not purely Euclidian problems in human computations, but involve emotional, autonomic, and cognitive constraints. They therefore require a level of parallel processing not possible in a macroscopic system to complete the algorithm within a brief period of time. A microscopic neurobiological model emphasizing the computational power of excited atoms within the neuronal membrane is presented as an alternative to classical connectionist approaches. The evolution of the system is viewed in terms of specific natural selection pressures driving satisfying computations toward global optimization. The relationship of microscopic computation to the nature of consciousness is examined, and possible mathematical models as a basis for simulation studies are briefly discussed.

Efficient local search with search space smoothing: a case study of the traveling salesman problem (TSP)

Local search is very efficient to solve combinatorial optimization problems. Due to the rugged terrain surface of the search space, it often gets stuck at a locally optimum configuration. In this paper, we give a local search method with a search space smoothing technique. It is capable of smoothing the rugged terrain surface of the search space. Any conventional heuristic search algorithm can be used in conjunction with this smoothing method. In a parameter space, by altering the shape of the objective function, the original problem instance is transformed into a series of gradually simplified problem instances with smoother terrain surfaces. Using an existing local search algorithm, an instance with the simplest terrain structure is solved first, the original problem instance with more complicated terrain structure is solved last, and the solutions of the simplified problem instances are used to guide the search of more complicated ones. A case study of using such technique to solve the traveling salesman problem (TSP) is described. We tested this method with numerous randomly generated TSP instances. We found that it has significantly improved the performance of existing heuristic search algorithms. >

Effective neural algorithms for the traveling salesman problem

Recently, several researchers have tried to solve the Traveling Salesman Problem (TSP) using the Hopfield model. Unfortunately, the results obtained so far are not good enough compared with the results obtained by conventional heuristic algorithms. This article presents two new neural algorithms. The first one uses Hopfield's neural network, and is a neural implementation of the Karp and Steele algorithm. Based on a generalized neural network, the second neural algorithm improves the first one by adaptively changing the neural network and thus the optimization function.On 40 TSP instances with random distance matrices, the Neural Algorithm 2 is better than Neural Algorithm 1 with confidence level α = 0.05, and much better than the 2-OPT (30 runs) and the Lin and Kernighan algorithm (30 runs).For random planar instances, Neural Algorithm 2 provides excellent initial solutions for iterative improvement algorithms, such as 2-OPT or Lin and Kernighan algorithm. For example, on 40 instances (each having 100 cities), the results of Neural Algorithm 2 improved by 2-OPT are better by at least 1.1% (with confidence level α = 0.0069) than the results produced by 2-OPT with 25 random initial solutions. If the results are postprocessed by the Lin and Kernighan algorithm, the Neural Algorithm 2 also outperforms the Lin and Kernighan algorithm with 20 random initial solutions (for each problem instance). Furthermore, the Neural Algorithm 2 scales up better than Lin and Kernighan algorithm (30 runs).

Analyzing the Held-Karp tsp bound: a monotonicity property with application

In their 1971 paper on the travelling salesman problem and minimum spanning trees, Held and Karp showed that finding an optimally weighted 1-tree is equivalent to solving a linear program for the traveling salesman problem (TSP) with only node-degree constraints and subtour elimination constraints. In this paper we show that the Held-Karp 1-trees have a certain monotonicity property: given a particular instance of the symmetric TSP with triangle inequality, the cost of the minimum weighted 1-tree is monotonic with respect to the set of nodes included. As a consequence, we obtain an alternate proof of a result of Wolsey and show that linear programs with node-degree and subtour elimination constraints must have a cost at least 2 3 OPT where OPT is the cost of the optimum solution to the TSP instance.

A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’

Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8 k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3 k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.