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An exact algorithm for the minimum quartet tree cost problem

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Abstract

The minimum quartet tree cost (MQTC) problem is a graph combinatorial optimization problem where, given a set of \(n \ge 4\) data objects and their pairwise costs (or distances), one wants to construct an optimal tree from the \(3 \cdot {n \atopwithdelims ()4}\) quartet topologies on n, where optimality means that the sum of the costs of the embedded (or consistent) quartet topologies is minimal. The MQTC problem is the foundation of the quartet method of hierarchical clustering, a novel hierarchical clustering method for non tree-like (non-phylogeny) data in various domains, or for heterogeneous data across domains. The MQTC problem is NP-complete and some heuristics have been already proposed in the literature. The aim of this paper is to present a first exact solution approach for the MQTC problem. Although the algorithm is able to get exact solutions only for relatively small problem instances, due to the high problem complexity, it can be used as a benchmark for validating the performance of any heuristic proposed for the MQTC problem.

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Notes

  1. Note that by construction there is always one and only one path connecting two leaves of a full unrooted binary tree.

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Acknowledgements

The author Dr. Sergio Consoli wants to dedicate this work with deepest respect to the memory of Professor Kenneth Darby-Dowman, a great scientist, an excellent manager, the best supervisor, a wonderful person, a real friend. He is also particularly grateful to Eng. Lucia Cantone, Prof. Fabrizio Consoli, Prof. Pierpaolo Vivo, Prof. Diego Reforgiato Recupero, and Eng. Niccolo’ Nobile, for helpful guidance and support, inspiring discussions, and precious advices during the development of this research work.

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Authors and Affiliations

  1. Philips Research, High Tech Campus 34, 5656 AE, Eindhoven, The Netherlands

    Sergio Consoli, Jan Korst & Steffen Pauws

  2. Netherlands Comprehensive Cancer Organisation (IKNL), Zernikestraat 29, 5612 HZ, Eindhoven, The Netherlands

    Gijs Geleijnse

  3. TiCC, Tilburg University, Warandelaan 2, 5037 AB, Tilburg, The Netherlands

    Steffen Pauws

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  1. Sergio Consoli
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  2. Jan Korst
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  3. Gijs Geleijnse
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  4. Steffen Pauws
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Correspondence to Sergio Consoli.

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Consoli, S., Korst, J., Geleijnse, G. et al. An exact algorithm for the minimum quartet tree cost problem. 4OR-Q J Oper Res 17, 401–425 (2019). https://doi.org/10.1007/s10288-018-0394-2

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  • DOI: https://doi.org/10.1007/s10288-018-0394-2

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