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The Wheels of the Orthogonal Latin Squares Polytope: Classification and Valid Inequalities

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Abstract

A wheel in a graph G(V,E) is an induced subgraph consisting of an odd hole and an additional node connected to all nodes of the hole. In this paper, we study the wheels of the intersection graph of the Orthogonal Latin Squares polytope (P I ). Our work builds on structural properties of wheels which are used to categorise them into a number of collectively exhaustive classes. These classes give rise to families of inequalities that are valid for P I and facet-defining for its set-packing relaxation. The classification introduced allows us to establish the cardinality of the whole wheel class and determine the range of the coefficients of any variable included in a lifted wheel inequality. Finally, based on this classification, a constant-time recognition algorithm for wheel-inducing circulant matrices, is introduced.

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References

  • G. Appa, D. Magos, I. Mourtos, and J.C.M. Janssen, “Clique facets of the orthogonal Latin squares polytope,” Research Report: LSE-CDAM-2001-04, London, 2001 (url: http://www.cdam.lse.ac.uk/Reports/Files/cdam-2001-04.pdf).

  • G. Appa, D. Magos, and I. Mourtos, “Properties and classification of the wheels of the OLS polytope,” Research Report: LSE-CDAM-2003-04, London, 2003 (url: http://www.cdam.lse.ac.uk/Reports/Files/cdam-2003-04.pdf).

  • F. Barahona and A.R. Mahjoub, “Compositions of graphs and polyhedra II: Stable sets,” SIAM J. Discrete Math., vol. 7, pp. 359–371, 1994.

    MathSciNet  Google Scholar 

  • E. Cheng and W.H. Cunningham, “Wheel inequalities for stable set polytopes,” Math. Program., vol. 77, pp. 389–421, 1997.

    Article  MathSciNet  Google Scholar 

  • E. Cheng and S. de Vries, “Antiweb-wheel inequalities and their separation problems over the stable set polytopes,” Math. Program. vol. 92, pp. 153–175, 2002.

  • G.B. Dantzig, Linear Programming and Extensions, Princeton University Press: Princeton, NJ, 1963.

  • R. Euler, R.E. Burkard, and R. Grommes, “On Latin squares and the facial structure of related polytopes,” Discrete Math., vol. 62, pp. 155–181, 1986.

    Article  MathSciNet  Google Scholar 

  • M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization (2nd corr. edition), Springer: Berlin, 1993.

  • C.F. Laywine and G.L. Mullen, Discrete Mathematics Using Latin Squares, John Wiley & Sons: New York, 1998.

    Google Scholar 

  • M.W. Padberg, “On the facial structure of the set packing polyhedra,” Math. Program. vol. 5, pp. 199–215, 1973.

  • D.B. West, Introduction to Graph Theory (2nd ed.), Prentice-Hall: Englewood Cliffs, NJ, 2000.

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Authors and Affiliations

  1. Operational Research Department, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    G. Appa

  2. Department of Informatics, Technological Educational Institute of Athens, Ag. Spyridonos Str., 122 10, Egaleo, Greece

    D. Magos

  3. Department of Economics, University of Patras, Rio 26500, Patras, Greece

    I. Mourtos

Authors
  1. G. Appa
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  2. D. Magos
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  3. I. Mourtos
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Correspondence to G. Appa.

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Appa, G., Magos, D. & Mourtos, I. The Wheels of the Orthogonal Latin Squares Polytope: Classification and Valid Inequalities. J Comb Optim 10, 365–389 (2005). https://doi.org/10.1007/s10878-005-4924-4

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  • DOI: https://doi.org/10.1007/s10878-005-4924-4

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