TOP

, Volume 22, Issue 1, pp 384–396 | Cite as

On a binary distance model for the minimum linear arrangement problem

Original Paper

Abstract

The minimum linear arrangement problem consists of finding an embedding of the nodes of a graph on the line such that the sum of the resulting edge lengths is minimized. The problem is among the classical NP-hard optimization problems and there has been extensive research on exact and approximative algorithms. In this paper, we introduce a new model based on binary variables dijk that are equal to 1 if nodes i and j have distance k in the ordering. We analyze this model and point to connections and differences to a model using integer distance variables. Based on computational experiments, we argue that our model is worth further theoretical and practical investigation and that is has potentials yet to be examined.

Keywords

Linear arrangement problem Graph layout Integer programming 

Mathematics Subject Classification

90C10 90C27 90C57 

References

  1. Adolphson D (1977) Single machine job sequencing with precedence constraints. SIAM J Comput 6(1):40–54 CrossRefGoogle Scholar
  2. Amaral A, Letchford AN (2012) A polyhedral approach to the single row facility layout probles. Math Program, to appear Google Scholar
  3. Caprara A, Letchford A, Salazar-Gonzáles J (2011a) Decorous lower bounds for minimum linear arrangement. INFORMS J Comput 23:26–40 CrossRefGoogle Scholar
  4. Caprara A, Oswald M, Reinelt G, Schwarz R, Traversi E (2011b) Optimal linear arrangements using betweenness variables. Math Program Comput. doi:10.1007/s12532-011-0027-7 Google Scholar
  5. Chen P (1976) The entity-relationship model—toward a unified view of data. ACM Trans Database Syst 1:9–36 CrossRefGoogle Scholar
  6. Díaz J, Petit J, Spirakis P (1998) Heuristics for the MinLA problem: am empirical and theoretical analysis (extended abstract). Working paper Google Scholar
  7. Díaz J, Petit J, Serna M (2002) A survey on graph layout problems. ACM Comput Surv 34(3):313–356 CrossRefGoogle Scholar
  8. Duff I, Grimes R, Lewis J (1992) User’s guide for the Harwell–Boeing sparse matrix collection. Technical report TR/PA/92/86, CERFACS, Toulouse, France Google Scholar
  9. Even G, Naor J, Rao S, Schieber B (2000) Divide-and-conquer approximation algorithms via spreading metrics. J ACM 47(4):585–616 CrossRefGoogle Scholar
  10. Even S, Shiloach Y (1978) NP-completeness of several arrangements problems. Technical report, TR-43, The Technicon, Haifa Google Scholar
  11. Fernau H (2005) Parameterized algorithmics: a graph-theoretic approach. Habilitation thesis, University of Tübingen Google Scholar
  12. Fernau H (2008) Parameterized algorithmics for linear arrangement problems. Discrete Appl Math 156(17):3166–3177 CrossRefGoogle Scholar
  13. Gane C, Sarson T (1977) Structured systems analysis: tools and techniques, 1st edn. McDonnell Douglas Systems Integration Company Google Scholar
  14. Gutin G, Rafiey A, Szeider S, Yeo A (2007) The linear arrangement problem parametrized above guaranteed value. Theory Comput Syst 41(3):521–538 CrossRefGoogle Scholar
  15. Hanan M, Kurtzberg J (1972) A review of the placement and quadratic assignment problems. SIAM Rev 14(2):324–342 CrossRefGoogle Scholar
  16. Harper L (1964) Optimal assignments of numbers to vertices. SIAM J Appl Math 12(1):131–135 CrossRefGoogle Scholar
  17. Hassin R, Rubinstein S (2000) Approximation algorithms for maximum linear arrangement. In: Algorithm theory—SWAT 2000. Lecture notes in computer science, vol 1851. Springer, Berlin, pp 633–643 Google Scholar
  18. Horton S (1997) The optimal linear arrangement problem: algorithms and approximation. PhD thesis, Georgia Institute of Technology, USA Google Scholar
  19. Ilog (2002) Cplex 8.1 Google Scholar
  20. Karp R (1993) Mapping the genome: some combinatorial problems arising in molecular biology. In: Proceedings of the twenty-fifth annual ACM symposium on theory of computing, pp 278–285 Google Scholar
  21. Koren Y, Harel D (2002) A multi-scale algorithm for the linear arrangement problem. In: Revised papers from the 28th international workshop on graph-theoretic concepts in computer science. Lecture notes in computer science, vol 2573, pp 296–309 CrossRefGoogle Scholar
  22. Liu W, Vannelli A (1995) Generating lower bounds for the linear arrangement problem. Discrete Appl Math 59(2):137–151 CrossRefGoogle Scholar
  23. Mitchison G, Durbin R (1986) Optimal numberings of an N×N array. SIAM J Algebr Discrete Methods 7(4):571–582 CrossRefGoogle Scholar
  24. Petit J (2001) Layout problems. PhD thesis, Universitat Politécnica de Catalunya Google Scholar
  25. Petit J (2003) Experiments on the minimum linear arrangement problem. ACM J Exp Algorithmics 8 Google Scholar
  26. Ravi R, Agrawal A, Klein P (1991) Ordering problems approximated: single-processor scheduling and interval graphs connection. Lect Notes Comput Sci 150:751–762 CrossRefGoogle Scholar
  27. Safro I, Ron D, Brandt A (2006) Graph minimum linear arrangement by multilevel weighted edge contractions. J Algorithms 60(1):24–41 CrossRefGoogle Scholar
  28. Schwarz R (2010) A branch-and-cut algorithm with betweenness variables for the linear arrangement problems. Master’s thesis, Universität Heidelberg Google Scholar
  29. Seitz H (2010) Contributions to the minimum linear arrangement problem. PhD thesis, Universität Heidelberg Google Scholar
  30. Serna M, Thilikos D (2005) Parameterized complexity for graph layout problems. Bull Eur Assoc Theor Comput Sci 86:41–65 Google Scholar
  31. Thienel S (1995) ABACUS: A Branch-And-CUt system. PhD thesis, Universtät zu Köln Google Scholar
  32. Vannelli A, Rowan G (1986) An eigenvector based approach for multistack VLSI layout. In: Proceedings of the midwest symposium on circuits and systems, vol 29, pp 435–439 Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2012

Authors and Affiliations

  1. 1.Institut für Informatik, Fakultät für Mathematik und InformatikUniversität HeidelbergHeidelbergGermany

Personalised recommendations