Edges versus circuits: a hierarchy of diameters in polyhedra
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Abstract
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.
Communicated by: M. Joswig
Acknowledgements
We are grateful for the comments we receive from Raymond Hemmecke, Jon Lee, and Edward Kim regarding some of these constructions. We are also grateful to Jake Miller for his help during the crafting of this paper.
Funding
The first author was supported by the Humboldt Foundation. The second author was supported by an NSA grant. The third author acknowledges the support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München.
References
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© 2016 by Walter de Gruyter Berlin/Boston
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