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12
Geometric optimization and sums of algebraic functions
, 2009
"... We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we appl ..."
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We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions, and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimumarea hulls. 1
Maximum Overlap of Convex Polytopes under Translation
"... We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε>0, finds an overlap at least the optimum minus ε and reports the translati ..."
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Cited by 6 (1 self)
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We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε>0, finds an overlap at least the optimum minus ε and reports the translation realizing it. The running time is O(n ⌊d/2⌋+1 log d n) with probability at least 1 − n −O(1),whichcanbeimprovedtoO(n log 3.5 n)inR 3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the bigO constants in these bounds are independent of ε.
Aligning two convex figures to minimize area or perimeter
, 2009
"... Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to interse ..."
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Cited by 4 (2 self)
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Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact nearlinear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1 + ε)approximation in time O(ε −1/2 log n + ε −3/2 log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a ∈ {0, 1, 2} depending on the version of the problem.
Polynomial time algorithms for maximizing the intersection volume of polytopes
 Pacific Journal of Optimization
"... Abstract: Suppose that we have several polytopes in Rd and we can translate them without rotation. Here we consider the intersection maximization problem, which asks the positions of the polytopes which maximizes the volume of their intersection. In this paper, we address this problem, and show that ..."
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Abstract: Suppose that we have several polytopes in Rd and we can translate them without rotation. Here we consider the intersection maximization problem, which asks the positions of the polytopes which maximizes the volume of their intersection. In this paper, we address this problem, and show that the problem can be solved in oracle polynomial time by an ellipsoid method, exploiting two important facts. Namely, the objective function is a continuous piecewisepolynomial function and its dth root is concave. We further study the structure of the problem in depth for the two dimensional case, and propose an algorithm which solves it in O(n4) time for two nonconvex polygons, where n is the total number of vertices.
Overlap of Convex Polytopes under Rigid Motion
, 2012
"... We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overl ..."
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We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overlap is at least λ · max{P1, P2} for some given constant λ ∈ (0, 1].
A Scan Matching Method based on the Area Overlap of StarShaped Polygons
"... AbstractWe illustrate a method that performs scan matching by maximizing the intersection area of the scans. The intersection area is a robust parameter that is less prone to measurement errors with respect to alternative techniques. Furthermore, such technique does not require to associate each p ..."
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AbstractWe illustrate a method that performs scan matching by maximizing the intersection area of the scans. The intersection area is a robust parameter that is less prone to measurement errors with respect to alternative techniques. Furthermore, such technique does not require to associate each point of one scan to a point of the other one like in some popular algorithms. The relative pose that maximizes the overlap is estimated iteratively. Since the scans are represented by starshaped polygons due to visibility properties, their intersection can be computed using an efficient lineartime traversal of the vertices. Then, the relative pose is updated under the hypothesis that the combinatorics of intersection is left unchanged and the procedure is repeated until the scans are aligned with sufficient precision.
Randomness and Computation Joint Workshop “New Horizons in Computing ” and “Statistical Mechanical Approach to Probabilistic
"... On finding a guard that sees most and a shop that sells most 1 ..."
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"... was preceded by a oneday workshop entitled “CGAL Innovations and Applications: Robust Geometric ..."
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was preceded by a oneday workshop entitled “CGAL Innovations and Applications: Robust Geometric
Probabilistic Matching of polygons
, 2008
"... We analyze a probabilistic algorithm for matching plane compact sets with sufficiently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is ..."
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We analyze a probabilistic algorithm for matching plane compact sets with sufficiently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. We give a time bound that does not depend on the number of vertices in the case of polygons.
Scandinavian Thins on Top of Cake: New and improved algorithms for stacking and packing
"... We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest conve ..."
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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NPhard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.