The universality theorems on the classification problem of configuration varieties and convex polytopes varieties

Mnëv, Nikolai

doi:10.1007/bfb0082792

Cited by 233 publications

(221 citation statements)

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“…Extending results of Mnëv [16] by using techniques described in [3] one finds (see [1,Cor. 9.5.11]) that there is a polynomial (Karp-)reduction of the problem to decide whether a system of linear inequalities has an integral solution to problem (S).…”

Section: Remarksmentioning

confidence: 77%

Reconstructing a Simple Polytope from Its Graph

Kaibel

2003

Combinatorial Optimization — Eureka, You Shrink!

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Abstract. 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].

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Section: Remarksmentioning

confidence: 77%

Reconstructing a Simple Polytope from Its Graph

Kaibel

2003

Combinatorial Optimization — Eureka, You Shrink!

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“…Some of the traditional ETR results are actually universality theorem; for example, Mnëv [34] showed that any semi-algebraic set is stably equivalent to the realization space of a pseudoline arrangement. We do not want to define stable equivalence explicitly (see [39] for a detailed discussion), but roughly speaking it means that the two sets look very similar algebraically.…”

Section: Issues Of Precisionmentioning

confidence: 99%

“…Many computational problems in geometry, graph drawing and other areas can be shown decidable using the (existential) theory of the real numbers; this includes the rectilinear crossing number, the Steinitz problem, and finding a Nash equilibrium; what is less often realized-with some exceptions-is that the existential theory of the reals captures the computational complexity of many of these problems precisely: deciding the truth of a sentence in the existential theory of the reals is polynomial-time equivalent to finding the rectilinear crossing number problem [3], solving the Steinitz problem [34,4], finding a Nash Equilibrium [43], recognizing intersection graphs of convex sets and ellipses [42], recognizing unit disk graphs [32] and many other problems. 1 In this paper we try to further substantiate this claim by showing that some well-known Euclidean realizability problems have the same complexity.…”

Section: Introductionmentioning

confidence: 99%

Realizability of Graphs and Linkages

Schaefer

2012

Thirty Essays on Geometric Graph Theory

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We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers, ETR; we introduce the class ∃R that captures the computational complexity of ETR and many other problems. The graph realizability problem remains ∃R-complete if all edges have unit length, which implies that recognizing unit distance graphs is ∃R-complete. We also consider the problem for linkages: in a realization of a linkage vertices are allowed to overlap and lie on the interior of edges. Linkage realizability is ∃R-complete and remains so if all edges have unit length. A linkage is called rigid if any slight perturbation of its vertices which does not break the linkage (i.e. keeps edge-lengths the same) is the result of a rigid motion of the plane. Testing whether a configuration is not rigid is ∃R-complete.

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“…(In fact, universality theorems [7,8] suggest, but do not prove, that it is probably at least NP-hard and PSPACE-hard, if not harder.) Thus, it is interesting to find efficiently computable subclasses of graphs which contain the graphs of simple polytopes.…”

Section: Polynomially Computable Classes Of Almost-polytopal Graphsmentioning

confidence: 99%

Finding a Simple Polytope from Its Graph in Polynomial Time

Friedman

2008

Discrete Comput Geom

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We show that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time. Our proof relies on a primal-dual characterization (by Joswig, Kaibel, and Körner in Israel J. Math. 129:109-118, 2002) and a linear program, with an exponential number of constraints, which can be used to construct the solution and can be solved in polynomial time. We show that this allows one to characterize the face lattice of the polytope, via a simple face recognition algorithm. In addition, we use this linear program to construct several interesting polynomial-time computable sets of graphs which may be of independent interest.

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The universality theorems on the classification problem of configuration varieties and convex polytopes varieties

Abstract: The results which we present here form the part of guiding by A.M.

Cited by 233 publications

References 2 publications

Reconstructing a Simple Polytope from Its Graph

Reconstructing a Simple Polytope from Its Graph

Realizability of Graphs and Linkages

Finding a Simple Polytope from Its Graph in Polynomial Time

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