An Oracle-Based, Output-Sensitive Algorithm for Projections of Resultant Polytopes

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

2016

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It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination.We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.

Section: Sparse Implicitizationmentioning

confidence: 99%

Compact Formulae in Sparse Elimination

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

2016

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“…[3]. Sparse implicitization relies on computing the Newton polytope of the sparse resultant or its orthogonal projection along a given direction [6], implemented in ResPol 2 .…”

Section: Previous Workmentioning

confidence: 99%

“…Then the implicit support is a subset of the set of lattice points contained in the predicted polytope, modulo the Minkoswki summand. For computing Q we employ [6] and software ResPol.…”

Section: Implicitization By Support Pre-dictionmentioning

confidence: 99%

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Minkowski Decomposition and Geometric Predicates in Sparse Implicitization

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

Konaxis

Zafeirakopoulos

2015

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Based on the computation of a polytope Q, called the predicted polytope, containing the Newton polytope P of the implicit equation, implicitization of a parametric hypersurface is reduced to computing the nullspace of a numeric matrix. Polytope Q may contain P as a Minkowski summand, thus jeopardizing the efficiency of sparse implicitization. Our contribution is twofold. On one hand we tackle the aforementioned issue in the case of 2D curves and 3D surfaces by Minkowski decomposing Q thus detecting the Minkowski summand relevant to implicitization: we design and implement in Sage a new, public domain, practical, potentially generalizable and worst-case optimal algorithm for Minkowski decomposition in 3D based on integer linear programming. On the other hand, we formulate basic geometric predicates, namely membership and sidedness for given query points, as rank computations on the interpolation matrix, thus avoiding to expand the implicit polynomial. This approach is implemented in Maple.

“…In this paper we study the case where a polytope P is given by an implicit representation, where the only access to P is a black box subroutine (oracle) that solves the LP problem on P for a given vector c. Then, we say that P is given by an optimization, or LP oracle. Given such an oracle, the entire polytope can be reconstructed, and both V-and H-representations can be found using the Beneath-Beyond method [EFKP13,Hug06], although not in total polynomial-time.…”

Section: Introductionmentioning

confidence: 99%

Efficient edge-skeleton computation for polytopes defined by oracles

Journal of Symbolic Computation

Fisikopoulos

Gärtner

2016

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In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify problems (and polytope representations) for which total polynomial-time algorithms can be obtained. We offer the first total polynomial-time algorithm for computing the edge-skeleton (including vertex enumeration) of a polytope given by an optimization or separation oracle, where we are also given a superset of its edge directions. We also offer a space-efficient variant of our algorithm by employing reverse search. All complexity bounds refer to the (oracle) Turing machine model. There is a number of polytope classes naturally defined by oracles; for some of them neither vertex nor facet representation is obvious. We consider two main applications, where we obtain (weakly) total polynomial-time algorithms: Signed Minkowski sums of convex polytopes, where polytopes can be subtracted provided the signed sum is a convex polytope, and computation of secondary, resultant, and discriminant polytopes. Further applications include convex combinatorial optimization and convex integer programming, where we offer a new approach, thus removing the complexity's exponential dependence in the dimension.