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Complexity and algorithms for the traveling salesman problem and the assignment problem of second order

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Abstract

We introduce two new combinatorial optimization problems, which are gener-alizations of the Traveling Salesman Problem (TSP) and the Assignment Problem (AP) and which we call Traveling Salesman Problem of Second Order (TSP2) and Assignment Problem of Second Order (AP2). TSP2 is motivated by an important ap-plication in bioinformatics, especially the Permuted Variable Length Markov model. While TSP2 is trivially N P-hard, we show the N P-hardness of AP2 by a reduc-tion from SAT. We propose seven elementary heuristics for the TSP2, some of which are generalizations of similar algorithms for the Traveling Salesman Problem, some of which are new ideas. Furthermore we give four exact algorithms for the TSP2, namely a Branch-and-Bound (BnB) algorithm, an Integer Programming (IP) algorithm, a Branch-and-Cut (BnC) algorithm and an algorithm based on a polynomial reduc-tion to the original TSP (TSP-R). Finally we experimentally compare the algorithms for many different random instances and real instances from the already mentioned application in bioinformatics. Our experiments show that for real instances most heuristics lead to optimal or almost-optimal solutions. For both, random and real classes, our most sophisticated exact approach BnC is the leading algorithm. In par-ticular, the BnC algorithm is able to solve real instances up to size 80 in reasonable time, proving the applicability of this approach.

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Many short DNA motifs, such as transcription factor binding sites (TFBS) and splice sites, exhibit strong local as well as nonlocal dependence. We introduce permuted variable length Markov models (PVLMM) which could capture the potentially important dependencies among positions and apply them to the problem of detecting splice and TFB sites. They have been satisfactory from the viewpoint of prediction performance and also give ready biological interpretations of the sequence dependence observed. The issue of model selection is also studied.
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In this paper we further investigate tour construction algorithms for the Asymmetric Traveling Salesman Problem (ATSP). In [1] we introduced a new algorithm, called Contract-or-Patch (COP). We have tested the algorithm together with other well-known and new heuristics on a variety of families of ATSP instances. In our study, COP has demonstrated good performance, clearly outperforming all other algorithms on robustness. It has either produced the shortest tours or came close to the leader on each of the seven families tested, while each of the remaining algorithms failed on at least two families of instances. In this paper we introduce three new variants of the COP algorithm, and perform an extensive computational study of the original as well as new versions of the algorithm on a variety of ATSP instances. We also study the inuence of the threshold parameter on the quality of tours produced by COP. We conclude the study by recommending one of the new versions of COP as ...
Molitor: Tolerances Applied in Combinatorial Optimiza-tion
  • B Goldengorin
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B. Goldengorin, G. J¨ ager, P. Molitor: Tolerances Applied in Combinatorial Optimiza-tion. J. Comput. Sci. 2(9), 716-734, 2006.
Bertsekas: A New Algorithm for the Assignment Problem
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Tolerance-Based Greedy Algorithms for the Traveling Salesman Problem Chapter 5 in: Mathematical Programming and Game Theory for Decision Making World Scientific
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D. Ghosh, B. Goldengorin, G. Gutin, G. Jäger: Tolerance-Based Greedy Algorithms for the Traveling Salesman Problem. Chapter 5 in: Mathematical Programming and Game Theory for Decision Making. S.K. Neogy, R.B. Bapat, A.K. Das, T. Parthasarathy (Eds.). World Scientific, New Jersey, 47-59, 2008.
University of Halle-Wittenberg, Chair for Bioinformatics, and Leibniz Institute of Plant Genetics and Crop Plant Research in Gatersleben
  • I Grosse
  • J Keilwagen
I. Grosse, J. Keilwagen, University of Halle-Wittenberg, Chair for Bioinformatics, and Leibniz Institute of Plant Genetics and Crop Plant Research in Gatersleben. Private communication.
Chair for Bioinformatics, and Leibniz Institute of Plant Genetics and Crop Plant Research in Gatersleben
  • I Grosse
  • J Keilwagen
I. Grosse, J. Keilwagen, University of Halle-Wittenberg, Chair for Bioinformatics, and Leibniz Institute of Plant Genetics and Crop Plant Research in Gatersleben. Private communication.