Calculus of Variations and Its Applications by Graves L. (ed.)

By Graves L. (ed.)

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A typical local-search algorithm follows the hillclimbing strategy of only moving to a neighboring solution if it is better than the one we have currently. In simulated annealing heuristics, this is relaxed to allow the algorithm to accept with a certain probability a neighbor that is worse than the current solution. At the start of the algorithm the probability of acceptance is high, but it is gradually decreased as the run progresses. The idea is to allow the algorithm to jump over to a better hill before switching to a steady climb.

The first angle suggests the obvious improvement of picking up Mt. Pulaski on the way from Springfield to Pekin. 13. The remaining acute angle around Shelbyville suggests the following two-step improvement: insert Decatur between Clinton and Monticello, then visit Sullivan before Shelbyville, after leaving Paris. 14, which is in fact the optimal tour. This ability to visualize tours and to easily manipulate them by hand has certainly contributed to the widespread appeal of the problem, making the study of the TSP accessible to anyone with a pencil and a clean sheet of paper.

Since good has many meanings, these days most researchers use the term polynomial-time algorithm to mean good in the sense of Edmonds. 4, it is easy to see why polynomial-time bounds are desirable; they exhibit modest growth with increasing problem size. 4 Values for n3 and n2 2n . n n3 n2 2n 5 125 800 10 1,000 102,400 20 8,000 419,430,400 40 64,000 1,759,218,604,441,600 there are some immediate objections to this general classification. First, one could hardly argue that an O(n1,000 ) algorithm is good in a practical sense.

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