  Blind and Mani (1987) 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 (1988) 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 vertex-facet 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 Koerner (2001).
