Skip to main content
SpringerLink
Log in

MIP model-based heuristics for the minimum weighted tree reconstruction problem

  • Original Paper
  • Published:
Operational Research Aims and scope Submit manuscript

Abstract

We consider the minimum weighted tree reconstruction (MWTR) problem and two matheuristic methods to obtain optimal or near-optimal solutions: the Feasibility Pump heuristic and the Local Branching heuristic. These matheuristics are based on a Mixed Integer Programming model used to find feasible solutions. We discuss the applicability and effectiveness of the matheuristics to obtain solutions to the MWTR problem. The purpose of the MWTR problem is to find a minimum weighted tree connecting a set of leaves in such a way that the length of the path between each pair of leaves is greater than or equal to a given distance between the considered pair of leaves. The Feasibility Pump matheuristic starts with the Linear Programming solution, iteratively fixes the values of some variables and solves the corresponding problem until a feasible solution is achieved. The Local Branching matheuristic, in its turn, improves a feasible solution by using a local search. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that these matheuristics are quite effective in finding feasible solutions and present small gap values. Each matheuristic can be used independently; however, the best results are obtained when used together. For instances of the problem having up to 17 leaves, the feasible solution obtained by the Feasibility Pump heuristic is improved by the Local Branching heuristic. Noticeably, when comparing with existing based models processes that solve instances having up to 15 leaves, this achievement of the matheuristic increases the size of solved instances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Achterberg T, Berthold T (2007) Improving the feasibility pump. Discrete Optim 4(1):77–86

    Article  Google Scholar 

  • Bertacco L, Fischetti M, Lodi A (2007) A feasibility pump heuristic for general mixed-integer problems. Discrete Optim 4(1):63–76

    Article  Google Scholar 

  • Bhamidi S, Rajagopal R, Roch S (2010) Network delay inference from additive metrics. Random Struct Algorithms 37(2):176–203

    Article  Google Scholar 

  • Boland NL, Eberhard AC, Engineer F, Tsoukalas A (2012) A new approach to the feasibility pump in mixed integer programming. SIAM J Optim 22(3):831–861

    Article  Google Scholar 

  • Byun S-S, Yoo C (2006) Reducing delivery delay in HRM tree. In: Gavrilova M, Gervasi O, Kumar V, Tan C, Taniar D, Laganà A, Mun Y, Choo H (eds) Computational science and its applications—ICCSA 2006, vol 3981. Lecture notes in computer science. Springer, Berlin, pp 1189–1198

    Chapter  Google Scholar 

  • Catanzaro D (2009) The minimum evolution problem: overview and classification. Networks 53(2):112–125

    Article  Google Scholar 

  • Catanzaro D, Aringhieri R, Summa MD, Pesenti R (2015) A branch-price-and-cut algorithm for the minimum evolution problem. Eur J Oper Res 244(3):753–765

    Article  Google Scholar 

  • Catanzaro D, Labbé M, Pesenti R, Salazar-Gonzaléz J-J (2009) Mathematical models to reconstruct phylogenetic trees under the minimum evolution criterion. Networks 53(2):126–140

    Article  Google Scholar 

  • Catanzaro D, Labbé M, Pesenti R, Salazar-Gonzaléz J-J (2012) The balanced minimum evolution problem. Inf J Comput 24(2):276–294

    Article  Google Scholar 

  • Cavalli-Sforza LL, Edwards AW (1967) Phylogenetic analysis. Models Estim Proc Evolut 21(3):550–570

    Google Scholar 

  • Chung F, Garrett M, Graham R, Shallcross D (2001) Distance realization problems with applications to internet tomography. J Comput Syst Sci 63(3):432–448

    Article  Google Scholar 

  • Coates M, Rabbat M, Nowak R (2003) Merging logical topologies using end-to-end measurements. In: Proceedings of the 3rd ACM SIGCOMM conference on internet measurement, IMC ’03, pp 192–203

  • Corter JE, Tversky A (1986) Extended similarity trees. Psychometrika 51(3):429–451

    Article  Google Scholar 

  • Cunningham JP (1980) Trees as memory representations for simple visual patterns. Memory Cogn 8(6):593–605

    Article  Google Scholar 

  • Day WH (1987) Computational complexity of inferring phylogenies from dissimilarity matrices. Bull Math Biol 49(4):461–467

    Article  Google Scholar 

  • De Soete G (1983) A least squares algorithm for fitting additive trees to proximity data. Psychometrika 48(4):621–626

    Article  Google Scholar 

  • Desper R, Gascuel O (2004) Theoretical foundation of the balanced minimum evolution method of phylogenetic inference and its relationship to weighted least-squares tree fitting. Mol Biol Evolut 21(3):587–598

    Article  Google Scholar 

  • Dias Z, Rocha A, Goldenstein S (2012) Image phylogeny by minimal spanning trees. IEEE Trans Inf Foren Secur 7(2):774–788

    Article  Google Scholar 

  • Donnet B, Friedman T (2007) Internet topology discovery: a survey. IEEE Commun Surv 9(4):2–14

    Article  Google Scholar 

  • Farach M, Kannan S, Warnow T (1995) A robust model for finding optimal evolutionary trees. Algorithmica 13(1):155–179

    Article  Google Scholar 

  • Felsenstein J (2004) Distance matrix methods. In Inferring Phylogenies, chapter 11, pp 147–175. Sinauer Associates, Inc., Sunderland, Massachusetts

  • Fiorini S, Joret G (2012) Approximating the balanced minimum evolution problem. Oper Res Lett 40(1):31–35

    Article  Google Scholar 

  • Fischetti M, Glover F, Lodi A (2005) The feasibility pump. Math Programm 104(1):91–104

    Article  Google Scholar 

  • Fischetti M, Lodi A (2003) Local branching. Math Programm 98(1–3):23–47

    Article  Google Scholar 

  • Fortz B, Oliveira O, Requejo C (2017) Compact mixed integer linear programming models to the minimum weighted tree reconstruction problem. Eur J Oper Res 256(1):242–251

    Article  Google Scholar 

  • Gilmore G, Hersh H, Caramazza A, Griffin J (1979) Multidimensional letter similarity derived from recognition errors. Percep Psychophys 25(5):425–431

    Article  Google Scholar 

  • Haddadi H, Rio M, Iannaccone G, Moore A, Mortier R (2008) Network topologies: inference, modelling and generation. IEEE Commun Surv 10(2):48–69

    Article  Google Scholar 

  • Huson D, Scornavacca C (2011) A survey of combinatorial methods for phylogenetic networks. Genome Biol Evol 3:23–35

    Article  Google Scholar 

  • Kumar S (1996) A stepwise algorithm for finding minimum evolution trees. Mol Biol Evol 13(4):584–593

    Article  Google Scholar 

  • Ni J, Xie H, Tatikonda S, Yang Y.R (2008) Network routing topology inference from end-to-end measurements. In: Proceedings of the IEEE INFOCOM 2008, the 27th Annual Joint Conference of the IEEE Computer and Communications Societies, pp 36–40

  • Parker D, Ram P (1996) The construction of Huffman codes is a submodular (“convex”) optimization problem over a lattice of binary trees. SIAM J Comput 28(5):1875–1905

    Article  Google Scholar 

  • Pauplin Y (2000) Direct calculation of a tree length using a distance matrix. J Mol Evol 51:41–47

    Article  Google Scholar 

  • Saitou N, Nei M (1987) The neighbor-joining method: a new method for reconstructing phylogenetic trees. Mol Biol Evol 4(4):406–425

    Google Scholar 

  • Sattath S, Tversky A (1977) Additive similarity trees. Psychometrika 42(3):319–345

    Article  Google Scholar 

Download references

Acknowledgments

Olga Fajarda was funded by the FCT under Grant SFRH/BD/76268/2011. This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020.

Author information

Authors and Affiliations

  1. Institute of Electronics and Informatics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193, Aveiro, Portugal

    Olga Fajarda

  2. University of Aveiro, 3810-193, Aveiro, Portugal

    Cristina Requejo

Authors
  1. Olga Fajarda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cristina Requejo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristina Requejo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fajarda, O., Requejo, C. MIP model-based heuristics for the minimum weighted tree reconstruction problem. Oper Res Int J 22, 2305–2342 (2022). https://doi.org/10.1007/s12351-020-00608-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12351-020-00608-z

Keywords

Search

Navigation