Skip to main content
SpringerLink
Log in

Cut and Flow Formulations for the Balanced Connected k-Partition Problem

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12176))

Included in the following conference series:

Abstract

For a fixed integer \(k\ge 2\), the balanced connected k-partition problem (\(\textsc {BCP}_k\)) consists in partitioning a graph into k mutually vertex-disjoint connected subgraphs of similar weight. More formally, given a connected graph G with nonnegative weights on the vertices, find a partition \(\{V_i\}_{i=1}^k\) of V(G) such that each class \(V_i\) induces a connected subgraph of G, and the weight of a class with the minimum weight is as large as possible. This problem, known to be \(\mathscr {N\!P}\)-hard, is used to model many applications arising in image processing, cluster analysis, operating systems and robotics. We propose an ILP and a MILP formulation for \(\textsc {BCP}_k\). The first one contains only binary variables and a potentially large number of constraints that can be separated in polynomial time. We also present polyhedral results on the polytope associated with this formulation, introduce new valid inequalities and design separation algorithms. The other formulation is based on flows and has a polynomial number of constraints and variables. Computational experiments show that our formulations achieve better results than the other formulations presented in the literature.

Research partially supported by grant #2015/11937-9, São Paulo Research Foundation (FAPESP). Miyazawa is supported by CNPq (Proc. 314366/2018-0 and 425340/2016-3) and FAPESP (Proc. 2016/01860-1). Moura is supported by FAPESP grants #2016/21250-3 and #2017/22611-2, CAPES, and Pró-Reitoria de Pesquisa da Universidade Federal de Minas Gerais. Ota is supported by CNPq. Wakabayashi is supported by CNPq (Proc. 306464/2016-0 and 423833/2018-9).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. de Aragão, M.P., Uchoa, E.: The \(\gamma \)-connected assignment problem. Eur. J. Oper. Res. 118(1), 127–138 (1999)

    MATH  Google Scholar 

  2. Assunção, T., Furtado, V.: A heuristic method for balanced graph partitioning: an application for the demarcation of preventive police patrol areas. In: Geffner, H., Prada, R., Machado Alexandre, I., David, N. (eds.) IBERAMIA 2008. LNCS (LNAI), vol. 5290, pp. 62–72. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-88309-8_7

    Chapter  Google Scholar 

  3. Barboza, E.U.: Problemas de classificação com restrições de conexidade flexibilizadas. Master’s thesis, Universidade Estadual de Campinas (1997)

    Google Scholar 

  4. Becker, R.I., Lari, I., Lucertini, M., Simeone, B.: Max-min partitioning of grid graphs into connected components. Networks 32(2), 115–125 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Becker, R.I., Schach, S.R., Perl, Y.: A shifting algorithm for min-max tree partitioning. J. ACM 29(1), 58–67 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borndörfer, R., Elijazyfer, Z., Schwartz, S.: Approximating balanced graph partitions. Technical report 19–25, ZIB, Takustr. 7, 14195 Berlin (2019)

    Google Scholar 

  7. Chataigner, F., Salgado, L.R.B., Wakabayashi, Y.: Approximation and inapproximability results on balanced connected partitions of graphs. Discrete Math. Theor. Comput. Sci. 9(1) (2007)

    Google Scholar 

  8. Chlebíková, J.: Approximating the maximally balanced connected partition problem in graphs. Inf. Process. Lett. 60(5), 225–230 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dezső, B., Jüttner, A., Kovács, P.: Lemon-an open source C++ graph template library. Electron. Notes Theor. Comput. Sci. 264(5), 23–45 (2011)

    Article  Google Scholar 

  10. Dyer, M., Frieze, A.: On the complexity of partitioning graphs into connected subgraphs. Discrete Appl. Math. 10(2), 139–153 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gleixner, A., Bastubbe, M., Eifler, L., et al.: The SCIP optimization suite 6.0. T. Report, optimization online, July 2018. http://www.optimization-online.org/DB_HTML/2018/07/6692.html

  12. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  14. Györi, E.: On division of graph to connected subgraphs. In: Combinatoris (Proceedings of Fifth Hungarian Colloquium, Koszthely, 1976), vol. I, Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam, New York, pp. 485–494 (1978)

    Google Scholar 

  15. Kawarabayashi, K., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. J. Combin. Theory Ser. B 102(2), 424–435 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lovász, L.: A homology theory for spanning tress of a graph. Acta Math. Acad. Sci. Hungarica 30, 241–251 (1977)

    Article  Google Scholar 

  17. Lucertini, M., Perl, Y., Simeone, B.: Most uniform path partitioning and its use in image processing. Discrete Appl. Math. 42(2), 227–256 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ma, J., Ma, S.: An O(\(k^2n^2\)) algorithm to find a \(k\)-partition in a \(k\)-connected graph. J. Comput. Sci. Technol. 9(1), 86–91 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Maravalle, M., Simeone, B., Naldini, R.: Clustering on trees. Comput. Stat. Data Anal. 24(2), 217–234 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matić, D.: A mixed integer linear programming model and variable neighborhood search for maximally balanced connected partition problem. Appl. Math. Comput. 237, 85–97 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Perl, Y., Schach, S.R.: Max-min tree partitioning. J. ACM 28(1), 5–15 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, B.Y.: Fully polynomial-time approximation schemes for the max-min connected partition problem on interval graphs. Discrete Math. Algorithms Appl. 04(01), 1250005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou, X., Wang, H., Ding, B., Hu, T., Shang, S.: Balanced connected task allocations for multi-robot systems: an exact flow-based integer program and an approximate tree-based genetic algorithm. Expert Syst. Appl. 116, 10–20 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computing, University of Campinas, Campinas, Brazil

    Flávio K. Miyazawa & Matheus J. Ota

  2. Department of Computer Science, Federal University of Minas Gerais, Belo Horizonte, Brazil

    Phablo F. S. Moura

  3. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil

    Yoshiko Wakabayashi

Authors
  1. Flávio K. Miyazawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Phablo F. S. Moura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matheus J. Ota
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yoshiko Wakabayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Phablo F. S. Moura .

Editor information

Editors and Affiliations

  1. CNRS and Université Clermont Auvergne, Aubière, France

    Mourad Baïou

  2. Département d'informatique et de recherche opérationnelle & CIRRELT, University of Montreal, Montréal, QC, Canada

    Bernard Gendron

  3. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA

    Oktay Günlük

  4. Laboratoire LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Miyazawa, F.K., Moura, P.F.S., Ota, M.J., Wakabayashi, Y. (2020). Cut and Flow Formulations for the Balanced Connected k-Partition Problem. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2020. Lecture Notes in Computer Science(), vol 12176. Springer, Cham. https://doi.org/10.1007/978-3-030-53262-8_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-53262-8_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-53261-1

  • Online ISBN: 978-3-030-53262-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation