Results 1 
8 of
8
How good are convex hull algorithms?
, 1996
"... A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are esse ..."
Abstract

Cited by 82 (8 self)
 Add to MetaCart
A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fatlattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.
The Vertex Set of a 0/1Polytope is Strongly PEnumerable
 Computational Geometry
, 1998
"... In this paper, we discuss the computational complexity of the following enumeration problem: Given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P . It is still an open question whether there exists an algorithm for listing all vertices in running t ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
In this paper, we discuss the computational complexity of the following enumeration problem: Given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P . It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of Penumerability introduced by Valiant [10]. The concept of strong Penumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1polytopes. Our implementation based on the commercial LP solver CPLEX 1 is superior to general vertex enumeration algorithms. We give an example how simplifications of...
Generating All Vertices of a Polyhedron Is Hard
 DISCRETE COMPUT GEOM (2008 ) 39 : 174–190 175
, 2008
"... We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other ne ..."
Abstract

Cited by 21 (6 self)
 Add to MetaCart
We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two wellknown generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
The Precision of Query Points as a Resource for Learning Convex Polytopes with Membership Queries
 In Proc. Conf. on Comp. Learning Theory
, 2000
"... We consider the problem of learning convex polytopes from membership queries only, where the learner is initially provided with a single interior point. The class of polytopes learnable in this setting turns out to be those whose vertices can be efficiently enumerated given their bounding hype ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We consider the problem of learning convex polytopes from membership queries only, where the learner is initially provided with a single interior point. The class of polytopes learnable in this setting turns out to be those whose vertices can be efficiently enumerated given their bounding hyperplanes. It is an open question whether in general one can enumerate the vertices of a given polytope in time polynomial in the number of vertices. In fact, we show that both problems are equivalent. We also give a querybased algorithm for the related problem of piecewise linear function regression. The bit complexity of the instances in our queries and the time complexity are polynomial in the bit complexity of the coefficients of the equations defining the bounding hyperplanes. This is consistent with prior established results showing that the weights, not the size, of a neural network determine the complexity of learning. As in previous positive results on learning convex...
The Negative Cycles Polyhedron and Hardness of Checking Some Polyhedral Properties
"... Given a graph G = (V, E) and a weight function on the edges w: E ↦→ R, we consider the polyhedron P(G, w) of negativeweight flows on G, and get a complete characterization of the vertices and extreme directions of P(G, w). Based on this characterization, and using a construction developed in [11], ..."
Abstract
 Add to MetaCart
Given a graph G = (V, E) and a weight function on the edges w: E ↦→ R, we consider the polyhedron P(G, w) of negativeweight flows on G, and get a complete characterization of the vertices and extreme directions of P(G, w). Based on this characterization, and using a construction developed in [11], we show that, unless P = NP, there is no output polynomialtime algorithm to generate all the vertices of a 0/1polyhedron. This strengthens the NPhardness result of [11] for non 0/1polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1polytopes [8]. As further applications, we show that it is NPhard to check if a given integral polyhedron is 0/1, or if a given polyhedron is halfintegral. Finally, we also show that it is NPhard to approximate the maximum support of a vertex a polyhedron in R n within a factor of 12/n.
Computational Geometry Theory and Applications The vertex set of a 0/1polytope is strongly 7)enumerable
, 1997
"... In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running ti ..."
Abstract
 Add to MetaCart
In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of 79enumerability introduced by Valiant (1979). The concept of strong 79enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1polytopes. Our implementation based on the commercial LP solver CPLEX 1 is superior to general vertex enumeration algorithms. We give an example how simplifications of our algorithm lead to efficient enumeration of combinatorial objects. © 1998 Elsevier Science B.V. All rights reserved.
unknown title
, 2007
"... Generating vertices of polyhedra and related monotone generation problems by ..."
Abstract
 Add to MetaCart
Generating vertices of polyhedra and related monotone generation problems by