Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order

We introduce a new combinatorial optimization problem, which is a generalization of the Traveling Salesman Problem (TSP) and which we call Traveling Salesman Problem of Second Order (2-TSP). It is motivated by an application in bioinformatics, especially the Permuted Variable Length Markov model. We propose seven elementary heuristics and two exact algorithms for the 2-TSP, some of which are generalizations of similar algorithms for the Asymmetric Traveling Salesman Problem (ATSP), some of which are new ideas. Finally we experimentally compare the algorithms for random instances and real instances from bioinformatics. Our experiments show that for the real instances most heuristics lead to optimum or almost-optimum solutions, and for the random instances the exact algorithms need less time than for the real instances.

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