, Volume 17, Issue 4, pp 401–425 | Cite as

An exact algorithm for the minimum quartet tree cost problem

  • Sergio ConsoliEmail author
  • Jan Korst
  • Gijs Geleijnse
  • Steffen Pauws
Research Paper


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.


Combinatorial optimization Quartet trees Minimum quartet tree cost Exact solution algorithms Cluster analysis Graphs 

Mathematics Subject Classification

90C27 05A05 05A15 62H30 68R10 05C30 92E10 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Philips ResearchEindhovenThe Netherlands
  2. 2.Netherlands Comprehensive Cancer Organisation (IKNL)EindhovenThe Netherlands
  3. 3.TiCCTilburg UniversityTilburgThe Netherlands

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