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PrimalDual Methods for Vertex and Facet Enumeration
 Discrete and Computational Geometry
, 1998
"... Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration) ..."
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Cited by 33 (7 self)
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Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primaldual ...
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 ..."
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Cited by 23 (0 self)
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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...
Exact volume computation for polytopes: a practical study
 in: Polytopes{combinatorics and computation (Oberwolfach
, 1997
"... ..."
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 ..."
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Cited by 21 (6 self)
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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... ..."
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Cited by 12 (1 self)
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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...
Incremental Convex Hull Algorithms Are Not Output Sensitive
, 1996
"... A polytope is the bounded intersection of a finite set of halfspaces of R d . Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V . The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalent ..."
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Cited by 10 (3 self)
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A polytope is the bounded intersection of a finite set of halfspaces of R d . Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V . The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalently by geometric duality) viceversa. Given an ordering v 1 : : : vn of the input vertices, after some initialization an incremental convex hull algorithm constructs halfspace descriptions Hn\Gammak : : : Hn where H i is the halfspace description of convf v 1 : : : v i g. Let m i denote jH i j, and let m denote mn . Let OE(d) denote d=d p d e \Gamma 1; in this paper we give families of polytopes for which mn\Gamma1 2 \Omega\Gamma m OE(d) ) for any ordering of the input. We also give a family of 0=1polytopes with a similar blowup in intermediate size. Since mn\Gamma1 is not bounded by any polynomial in m, n, and d, incremental convex hull algorithms cannot in any reasonable sense be con...
Computing the Face Lattice of a Polytope from its VertexFacet Incidences
, 2001
"... We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertexfacet incidences in time O(minfn; mg '), where n is the number of vertices, m is the number of facets, is the number of vertexfacet incidences, and ' is the total number of face ..."
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Cited by 9 (3 self)
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We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertexfacet incidences in time O(minfn; mg '), where n is the number of vertices, m is the number of facets, is the number of vertexfacet incidences, and ' is the total number of faces of P . This improves results of Fukuda and Rosta [4], who described an algorithm for enumerating all faces of a dpolytope P in O minfn; mg d ' 2 steps. For simple or simplicial dpolytopes our algorithm can be specialized to run in time O(d '). Furthermore,
The Lasso Problem and Uniqueness
, 2012
"... The lasso is a popular tool for sparse linear regression, especially for problems in which the number of variables p exceeds the number of observations n. But when p> n, the lasso criterion is not strictly convex, and hence it may not have a unique minimum. An important question is: when is the lass ..."
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Cited by 1 (0 self)
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The lasso is a popular tool for sparse linear regression, especially for problems in which the number of variables p exceeds the number of observations n. But when p> n, the lasso criterion is not strictly convex, and hence it may not have a unique minimum. An important question is: when is the lasso solution welldefined (unique)? We review results from the literature, which show that if the predictor variables are drawn from a continuous probability distribution, then there is a unique lasso solution with probability one, regardless of the sizes of n and p. We also show that this result extends easily to ℓ1 penalized minimization problems over a wide range of loss functions. A second important question is: how can we manage the case of nonuniqueness in lasso solutions? In light of the aforementioned result, this case really only arises when some of the predictor variables are discrete, or when some postprocessing has been performed on continuous predictor measurements. Though we certainly cannot claim to provide a complete answer to such a broad question, we do present progress towards understanding some aspects of nonuniqueness. First, we extend the LARS algorithm for computing the lasso solution path to cover the nonunique case, so that this path algorithm works for any predictor matrix. Next, we derive a simple method for computing the componentwise uncertainty in lasso solutions of any given problem instance, based on linear programming. Finally, we review results from the literature on some of the unifying properties of lasso solutions, and also point out particular forms of solutions that have distinctive properties. 1