Results 1  10
of
28
Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
, 2003
"... We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We me ..."
Abstract

Cited by 150 (14 self)
 Add to MetaCart
We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has smoothed complexity polynomial in the input size and the standard deviation of
New Lower Bounds for Convex Hull Problems in Odd Dimensions
 SIAM J. Comput
, 1996
"... We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result fo ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
We show that in the worst case, &Omega;(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasisimplicial nvertex polytope with &Omega;(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that ddimensional convex hulls can have &Omega;(n bd=2c ) facets, the previously best lower bound for these problems is only &Omega;(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is &lceil;d/2&rceil;hard, in the in the sense of Gajentaan and Overmars.
Smoothed Analysis of Termination of Linear Programming Algorithms
"... We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng
Topological obstructions for vertex numbers of Minkowski sums
, 2007
"... We show that for polytopes P1, P2,..., Pr ⊂ Rd, each having ni ≥ d + 1 vertices, the Minkowski sum P1 + P2 + · · · + Pr cannot achieve the maximum of ∏ i ni vertices if r ≥ d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d − 1 summands. T ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
We show that for polytopes P1, P2,..., Pr ⊂ Rd, each having ni ≥ d + 1 vertices, the Minkowski sum P1 + P2 + · · · + Pr cannot achieve the maximum of ∏ i ni vertices if r ≥ d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d − 1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen–type obstructions) as developed in Rörig, Sanyal, and Ziegler (2007).
Construction and analysis of projected deformed products
, 2007
"... We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly p ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly preserved ” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)polytope Q on n−1 vertices we construct a deformed ncube, whose projection to the last d coordinates yields a neighborly cubical dpolytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs), which have a projection to dspace that retains the complete ( ⌊ d 2 ⌋ − 1)skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the projected deformed products of polygons that were announced by Ziegler (2004), a family of 4polytopes whose “fatness ” gets arbitrarily close to 9. 1
Lower Bounds for Fundamental Geometric Problems
 IN 5TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA'97
, 1996
"... We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar question ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal. Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorit...
Extended Convex Hull
, 2000
"... In this paper we address the problem of computing a minimal Hrepresentation of the convex hull of the union of k Hpolytopes in R^d. Our method applies the reverse search algorithm to a shelling ordering of the facets of the convex hull. Efficient wrapping is done by projecting the polytopes onto t ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
In this paper we address the problem of computing a minimal Hrepresentation of the convex hull of the union of k Hpolytopes in R^d. Our method applies the reverse search algorithm to a shelling ordering of the facets of the convex hull. Efficient wrapping is done by projecting the polytopes onto the twodimensional space and solving a linear program. The resulting algorithm is polynomial in the sizes of input and output under the general position assumption.
On the Hardness and Smoothed Complexity of QuasiConcave Minimization
"... In this paper, we resolve the smoothed and approximative complexity of lowrank quasiconcave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasiconcave minimization. The analysis is based on a smoothed bound for the number of extr ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In this paper, we resolve the smoothed and approximative complexity of lowrank quasiconcave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasiconcave minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a kdimensional subspace, where k is the rank (informally, the dimension of nonconvexity) of the quasiconcave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size ρ, and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomialtime approximation scheme for lowrank quasiconcave minimization under broad conditions. In contrast with this, we prove log nhardness of approximation for general quasiconcave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasiconcave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto twodimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2planes in R n. In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higherdimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest: we show that the stochastic 2stage minimum spanning tree problem has a supermodular objective and that su
Extremal Properties for Dissections of Convex 3Polytopes
 Siam J. Discrete Math
, 1999
"... A dissection of a convex dpolytope is a partition of the polytope into dsimplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the nu ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
A dissection of a convex dpolytope is a partition of the polytope into dsimplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of dsimplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3polytope and analyze extremal size triangulations for specific nonsimplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial dcubes.
Two new bounds for the randomedge simplex algorithm
, 2008
"... We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for s ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial dcubes, the trivial upper bound of 2 d on the performance of RandomEdge can asymptotically be improved by any desired polynomial factor in d.