A simple way to tell a simple polytope from its graph
Journal of Combinatorial Theory Series A
From linear separability to unimodality: a hierarchy of pseudo-Boolean functions
SIAM Journal on Discrete Mathematics
The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Finding a Polytope from Its Graph in Polynomial Time
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
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Blind and Mani [2] proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai [15] found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertexfacet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Körner [12].