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On the Circuit Diameter Conjecture

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Abstract

From the point of view of optimization, a critical issue is relating the combinatorial diameter of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and Walkup (Acta Math 117:53–78, 1967), the Hirsch conjecture of an upper bound of \(f-d\) was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron \(U_4\) with eight facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos (Ann Math 176(1):383–412, 2012). We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the polyhedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equivalences in Klee and Walkup, the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra—in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture. These results offer some hope that the circuit version of the Hirsch conjecture may hold, even for unbounded polyhedra. A challenge in the circuit setting is that different realizations of polyhedra with the same combinatorial structure may have different diameters. We adapt the notion of simplicity to work with circuits in the form of \(\mathcal {C}\)-simple and wedge-simple polyhedra. We show that it suffices to consider such polyhedra for studying circuit analogues of the Hirsch conjecture.

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Notes

  1. 1.

    We thank the anonymous referee for the suggestion.

  2. 2.

    A preliminary version of this result appeared in the proceedings of Eurocomb 2015 [26].

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Acknowledgements

This research was partially supported by an NSERC Discovery Grant for T. Stephen, and an NSERC Postgraduate Scholarship-D for T. Yusun. All illustrations were produced using the GeoGebra software (http://www.geogebra.org, International GeoGebra Institute). We thank the anonymous referees, J. de Loera and E. Finhold for comments and encouragement.

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Correspondence to Tamon Stephen.

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Editor in Charge: János Pach

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Borgwardt, S., Stephen, T. & Yusun, T. On the Circuit Diameter Conjecture. Discrete Comput Geom 60, 558–587 (2018). https://doi.org/10.1007/s00454-018-9995-y

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Keywords

  • Circuit
  • Diameter
  • Polyhedra
  • Hirsch conjecture

Mathematics Subject Classification

  • 52B05
  • 90C05