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Cutting Planes and the Parameter Cutwidth

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Abstract

We introduce the parameter cutwidth for the Cutting Planes (CP) system of Gomory and Chvátal. We provide linear lower bounds on cutwidth for two simple polytopes. Considering CP as a propositional refutation system, one can see that the cutwidth of a CNF contradiction F is always bound above by the Resolution width of F. We provide an example proving that the converse fails: there is an F which has constant cutwidth, but has Resolution width Ω(n). Following a standard method for converting an FO sentence ψ, without finite models, into a sequence of CNFs, F ψ,n , we provide a classification theorem for CP based on the sum cutwidth plus rank. Specifically, the cutwidth + rank of F ψ,n is bound by a constant c (depending on ψ only) iff ψ has no (infinite) models. This result may be seen as a relative of various gap theorems extant in the literature.

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Notes

  1. It is conventional to consider vector spaces over the field ℝ, though one could equally consider ℚ.

  2. We omit functions and constants only for the sake of a clearer exposition; note that we may simulate constants in a single FO sentence with added outermost existential quantification on new variables replacing those constants.

  3. We resist calling it a gap theorem as the separation of constant and non-constant does not entail a gap.

  4. Note that, while we do not allow constants in our signature, we refer to those elements that have been mentioned in decision tree questions as constants. Also, we tend to discount the empty model. It is, therefore, possible to have ψ with no finite models and no outermost existential quantifier. In this case we may instantiate a single constant at the outset to get us going.

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Authors and Affiliations

  1. Department of Computer Science, University of Durham, Durham, UK

    Stefan Dantchev

  2. Équipe de Logique Mathématique, CNRS UMR 7056, Université Denis Diderot, Paris 7, France

    Barnaby Martin

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  1. Stefan Dantchev
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  2. Barnaby Martin
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Correspondence to Stefan Dantchev.

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Dantchev, S., Martin, B. Cutting Planes and the Parameter Cutwidth. Theory Comput Syst 51, 50–64 (2012). https://doi.org/10.1007/s00224-011-9373-0

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