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45
A Subexponential Bound for Linear Programming
 ALGORITHMICA
, 1996
"... We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorith ..."
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Cited by 166 (16 self)
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We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson’s linear programming algorithm, this gives an expected bound of O(d 2 n + e O( √ d ln d) The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in dspace, computing the distance of two nvertex (or nfacet) polytopes in dspace, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 93 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
HellyType Theorems and Geometric Transversals
 Handbook of Discrete and Computational Geometry, chapter 4
, 1997
"... INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., ..."
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Cited by 33 (3 self)
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INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., a point transversal. In Section 1 we highlight some of the more notable theorems related to Helly's Theorem and point transversals. Section 2 is devoted to geometric transversal theory. 4.1 HELLYTYPE THEOREMS In 1913, Eduard Helly proved the following theorem: Theorem 1 (Helly's Theorem) Let A be a finite family of at least d + 1 convex sets in R d . If every d + 1 members of A have a point in common, then there is a point common to all the members of A. The theorem also holds for infinite families
Rectilinear and Polygonal pPiercing and pCenter Problems
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axispa ..."
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Cited by 28 (1 self)
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We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex coriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) pcenter problem, in which we are given a set S of n points in the plane, and wish to find p axisparallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a lineartime solution for the rectilinear 3center problem (by showing that this problem can be formulated as an LPtype problem and by exhibiting a relation to Helly numbers). We give O(n log n...
Quasiconvex Analysis of Backtracking Algorithms
, 2003
"... We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to ..."
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Cited by 26 (2 self)
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We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to reduce the problem to that of solving univariate linear recurrences; show how to use quasiconvex programming to determine the weight function yielding the smallest upper bound; and prove that the resulting upper bounds are within a polynomial factor of the true asymptotics of the recurrence. We develop and implement a multiplegradient descent algorithm for the resulting quasiconvex programs, using a realnumber arithmetic package for guaranteed accuracy of the computed worst case time bounds.
Linear programming  randomization and abstract frameworks
 In Proc. 13th annu. Symp. on Theoretical Aspects of Computer Science (STACS
, 1996
"... Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial ’ algorithm which solves a linear program over d variables and n constraints with expected O(d 2 n + e O( d ..."
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Cited by 24 (9 self)
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Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial ’ algorithm which solves a linear program over d variables and n constraints with expected O(d 2 n + e O( d log d)) arithmetic operations. The bound relies on two algorithms by Clarkson, and the subexponential algorithms due to Kalai, and to Matouˇsek, Sharir & Welzl. frameworks like LPtype problems and abstract optimization problems (due to Gärtner) which allow the application of these algorithms to a number of nonlinear optimization problems (like polytope distance and smallest enclosing ball of points).
A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization
 Discr. Comput. Geometry
, 2000
"... Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then si ..."
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Cited by 22 (3 self)
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Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms. Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma) which holds in a general framework, yet has not been stated explicitly in the literature. In the more restricted but still general setting of LPtype problems, we prove tail estimates for the sampling lemma, giving Chernofftype bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple pricing, a heu...
Hadwiger and Hellytype theorems for disjoint unit spheres in R³
 IN PROC. 20TH ANN. SYMP. ON COMPUTATIONAL GEOMETRY
, 2005
"... Let S be an ordered set of disjoint unit spheres in R³. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every ..."
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Cited by 21 (14 self)
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Let S be an ordered set of disjoint unit spheres in R³. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S.
Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All Lptype Problems
, 2007
"... We show that a Simple Stochastic Game (SSG) can be formulated as an LPtype problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LPtype problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known ..."
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Cited by 16 (0 self)
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We show that a Simple Stochastic Game (SSG) can be formulated as an LPtype problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LPtype problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs. To the best of our knowledge, the LPtype framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LPtype framework in other fields, and for achieving subexponential algorithms.
Optimal Möbius Transformations for Information Visualization and Meshing
 Meshing, WADS 2001, Lecture Notes in Computer Science 2125
, 2001
"... . We give lineartime quasiconvex programming algorithms for ..."
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Cited by 15 (4 self)
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. We give lineartime quasiconvex programming algorithms for