Vissarion Fisikopoulos

An output-sensitive algorithm for computing projections of resultant polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6- or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.

Efficient Random-Walk Methods for Approximating Polytope Volume

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytopes volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problems geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B11, …, B15, whereas exact methods have only computed that of B10.

AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is alpha-n-1, where the input consists of alpha points in Z^n. Our approach is output-sensitive as it makes one oracle call per vertex and facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7-dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in <1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

Enumeration of 2-Level Polytopes

We propose the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d ≤ 6. Our approach is based on the notion of a simplicial core, that allows us to reduce the problem to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.

Faster geometric algorithms via dynamic determinant computation

Determinant computation is the core procedure in many important geometric algorithms, such as convex hull computations and point locations. As the dimension of the computation space grows, a higher percentage of the computation time is consumed by these predicates. In this paper we study the sequences of determinants that appear in geometric algorithms. We use dynamic determinant algorithms to speed-up the computation of each predicate by using information from previously computed predicates. We propose two dynamic determinant algorithms with quadratic complexity when employed in convex hull computations, and with linear complexity when used in point location problems. Moreover, we implement them and perform an experimental analysis. Our implementations outperform the state-of-the-art determinant and convex hull implementations in most of the tested scenarios, as well as giving a speed-up of 78 times in point location problems.

Two-Level Polytopes with a Prescribed Facet

A (convex) polytope is said to be 2-level if for every facet-defining direction of hyperplanes, its vertices can be covered with two hyperplanes of that direction. These polytopes are motivated by questions, e.g., in combinatorial optimization and communication complexity. We study 2-level polytopes with one prescribed facet. Based on new general findings about the structure of 2-level polytopes, we give a complete characterization of the 2-level polytopes with some facet isomorphic to a sequentially Hanner polytope, and improve the enumeration algorithm of Bohn et al. (ESA 2015). We obtain, for the first time, the complete list of d-dimensional 2-level polytopes up to affine equivalence for dimension \(d = 7\). As it turns out, geometric constructions that we call suspensions play a prominent role in both our theoretical and experimental results. This yields exciting new research questions on 2-level polytopes, which we state in the paper.

Efficient edge-skeleton computation for polytopes defined by oracles

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 complexitys exponential dependence in the dimension.

Practical Polytope Volume Approximation

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear halfspaces. We implement and evaluate randomized polynomial-time algorithms for accurately approximating the polytope’s volume in high dimensions (e.g., few hundreds) based onhit-and-run random walks. To carry out this efficiently, we experimentally correlate the effect of parameters, such as random walk length and number of sample points, with accuracy and runtime. Our method is based on Monte Carlo algorithms with guaranteed speed and provably high probability of success for arbitrarily high precision. We exploit the problem’s features in implementing a practical rounding procedure of polytopes, in computing only partial “generations” of random points, and in designing fast polytope boundary oracles. Our publicly available software is significantly faster than exact computation and more accurate than existing approximation methods. For illustration, volume approximations of Birkhoff polytopes B11,…,B15 are computed, in dimensions up to 196, whereas exact methods have only computed volumes of up to B10.

Combinatorics of 4-dimensional resultant polytopes

The Newton polytope of the resultant, or resultant polytope, characterizes the resultant polynomial more precisely than total degree. The combinatorics of resultant polytopes are known in the Sylvester case [Gelfand et al.90] and up to dimension 3 [Sturmfels 94]. We extend this work by studying the combinatorial characterization of 4-dimensional resultant polytopes, which show a greater diversity and involve computational and combinatorial challenges. In particular, our experiments, based on software respol for computing resultant polytopes, establish lower bounds on the maximal number of faces. By studying mixed subdivisions, we obtain tight upper bounds on the maximal number of facets and ridges, thus arriving at the following maximal f-vector: (22,66,66,22), i.e. vector of face cardinalities. Certain general features emerge, such as the symmetry of the maximal f-vector, which are intriguing but still under investigation. We establish a result of independent interest, namely that the f-vector is maximized when the input supports are sufficiently generic, namely full dimensional and without parallel edges. Lastly, we offer a classification result of all possible 4-dimensional resultant polytopes.

Enumeration of 2-level polytopes

A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for