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Linear Ordering Based MIP Formulations for the Vertex Separation or Pathwidth Problem

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Book cover Combinatorial Algorithms (IWOCA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10765))

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Abstract

We consider the task to compute the pathwidth of a graph which has been shown to be equivalent to the vertex separation problem. The latter is naturally modeled as a linear ordering problem w.r.t. the vertices of the graph. Mixed-integer programs proposed so far express linear orders using either position or set assignment variables. As we show, the lower bound on the pathwidth obtained when solving their linear programming relaxations is zero for any directed graph. We then present a new formulation based on conventional linear ordering variables and a slightly different perspective on the problem that sustains stronger lower bounds. An experimental evaluation of three mixed-integer programs, each representing one of the different modeling schemes, displays their potentials and limitations when used to solve the problem to optimality.

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Notes

  1. 1.

    SageMath is an open-source mathematics library http://www.sagemath.org.

  2. 2.

    The model in [19] can be implemented such that its relaxation yields non-zero bounds for some graphs, but it remains of too excessive size and inferior to \(\textit{MIP}_P\) in practice.

  3. 3.

    IBM ILOG CPLEX Optimization Studio is a proprietary LP and MIP solver.

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Authors and Affiliations

  1. Department of Computer Science, Universität zu Köln, Köln, Germany

    Sven Mallach

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  1. Sven Mallach
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Correspondence to Sven Mallach .

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Editors and Affiliations

  1. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Ljiljana Brankovic

  2. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Joe Ryan

  3. McMaster University, Hamilton, Ontario, Canada

    William F. Smyth

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Mallach, S. (2018). Linear Ordering Based MIP Formulations for the Vertex Separation or Pathwidth Problem. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_27

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  • DOI: https://doi.org/10.1007/978-3-319-78825-8_27

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