Low-dimensional linear programming and LP-type problems
Dynamic three-dimensional linear programming.
D. Eppstein.
Tech. Rep. 91-53, ICS, UCI, 1991.
32nd IEEE Symp. Foundations of Comp. Sci., San Juan, Puerto Rico, 1991, pp. 488–494.
ORSA J. Computing 4: 360–368, 1992 (special issue on computational geometry).Uses Dobkin-Kirkpatrick hierarchies to perform linear programming queries in the intersection of several convex polyhedra. By maintaining a collection of halfspaces as several subsets, represented by polyhedra, this leads to algorithms for a dynamic linear program in which updates change the set of constraints. The fully dynamic results have largely been subsumed by Agarwal and Matoušek, but this paper also includes polylog time results for semi-online problems, and uses them to give a fast randomized algorithm for the planar 2-center problem (later improved by various authors, most recently in "Faster Construction of Planar Two-Centers", which re-uses the data structures described here).
Approximating center points with iterated Radon points.
K. Clarkson, D. Eppstein, G.L. Miller, C. Sturtivant, and S.-H. Teng.
9th ACM Symp. Comp. Geom., San Diego, 1993, pp. 91–98.
Int. J. Comp. Geom. & Appl. 6 (3): 357–377, 1996.Given a collection of n sites, a center point is a point (not necessarily a site) such that no hyperplane through the centerpoint partitions the collection into a very small and a very large subset. Center points have been used by Teng and others as a key step in the construction of geometric separators. One can find a point with this property by choosing a random sample of the collection and applying linear programming, but the complexity of that method grows exponentially with the dimension. This paper proposes an alternate method that produces lower quality approximations (in terms of the size of the worst hyperplane partition) but takes time polynomial in both n and d.
The centroid of points with approximate weights.
M. Bern, D. Eppstein, L. J. Guibas, J. Hershberger, S. Suri, and J. Wolter.
3rd Eur. Symp. Algorithms, Corfu, 1995.
Springer, Lecture Notes in Comp. Sci. 979, 1995, pp. 460–472.Given a set of points with weights that are not known precisely, but are known to fall within some range, considers the possible weighted centroids arising from different choices of weights in each range. The combinatorics of this problem are closely connected with those of zonotopes.
Choosing subsets with maximum weighted average.
D. Eppstein and D. S. Hirschberg.
Tech. Rep. 95-12, ICS, UCI, 1995.
5th MSI Worksh. on Computational Geometry, 1995, pp. 7–8.
J. Algorithms 24: 177–193, 1997.Uses geometric optimization techniques to find, among n weighted values, the k to drop so as to maximize the weighted average of the remaining values. The feasibility test for the corresponding decision problem involves k-sets in a dual line arrangement.
Optimal point placement for mesh smoothing.
N. Amenta, M. Bern, and D. Eppstein.
8th ACM-SIAM Symp. Discrete Algorithms, New Orleans, 1997, pp. 528–537.
Symp. Computational Geometry Approaches to Mesh Generation, SIAM 45th Anniversary Mtg., Stanford, 1997.
arXiv:cs.CG/9809081.
J. Algorithms 30: 302–322, 1999 (special issue for SODA 1997).We study finite element mesh smoothing problems in which we move vertex locations to optimize the shapes of nearby triangles. Many such problems can be solved in linear time using generalized linear programming; we also give efficient algorithms for some non-LP-type mesh smoothing problems. One lemma may be of independent interest: the locus of points in Rd from which a d-1 dimensional convex set subtends a given solid angle is convex.
Setting parameters by example.
D. Eppstein.
arXiv:cs.DS/9907001.
40th IEEE Symp. Foundations of Comp. Sci., 1999, pp. 309–318.
SIAM J. Computing 32 (3): 643–653, 2003.We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We use low-dimensional linear programming and geometric sampling techniques to solve such problems for minimum spanning trees, shortest paths, and other optimal subgraph problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.
Optimal Möbius transformations for information visualization and meshing.
M. Bern and D. Eppstein.
arXiv:cs.CG/0101006.
7th Worksh. Algorithms and Data Structures, Providence, Rhode Island, 2001.
Springer, Lecture Notes in Comp. Sci. 2125, 2001, pp. 14–25.We give linear-time quasiconvex programming algorithms for finding a Möbius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, and element size control in conformal structured mesh generation.
Optimization over zonotopes and training support vector machines.
M. Bern and D. Eppstein.
arXiv:cs.CG/0105017.
7th Worksh. Algorithms and Data Structures, Providence, Rhode Island, 2001.
Springer, Lecture Notes in Comp. Sci. 2125, 2001, pp. 111–121.We use the ellipsoid method to develop (theoretically) efficient algorithms for optimizing linear functions on intersections of zonotopes, and show how to apply this to train soft-margin support vector classifiers.
Optimized color gamuts for tiled displays.
M. Bern and D. Eppstein.
arXiv:cs.CG/0212007.
19th ACM Symp. Comp. Geom., San Diego, 2003, pp. 274–281.We consider the problem of finding a large color space that can be generated by all units in multi-projector tiled display systems. Viewing the problem geometrically as one of finding a large parallelepiped within the intersection of multiple parallelepipeds, and using colorimetric principles to define a volume-based objective function for comparing feasible solutions, we develop an algorithm for finding the optimal gamut in time O(n3), where n denotes the number of projectors in the system. We also discuss more efficient quasiconvex programming algorithms for alternative objective functions based on maximizing the quality of the color space extrema.
Quasiconvex analysis of backtracking algorithms.
D. Eppstein.
arXiv:cs.DS/0304018.
15th ACM-SIAM Symp. Discrete Algorithms, New Orleans, 2004, pp. 781–790.
ACM Trans. Algorithms 2 (4): 492–509 (special issue for SODA 2004), 2006.We consider a class of multivariate recurrences frequently arising in the worst case analysis of Davis-Putnam-style exponential time backtracking algorithms for NP-hard 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 multiple-gradient descent algorithm for the resulting quasiconvex programs, using a real-number arithmetic package for guaranteed accuracy of the computed worst case time bounds.
The journal version uses the longer title "Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms".
Quasiconvex programming.
D. Eppstein.
Invited talk at DIMACS Worksh. on Geometric Optimization, New Brunswick, NJ, 2003.
Plenary talk at ALGO 2004, Bergen, Norway, 2004.
arXiv:cs.CG/0412046.
Combinatorial and Computational Geometry, Goodman, Pach, and Welzl, eds., MSRI Publications 52, 2005, pp. 287–331.Defines quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. Surveys algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.
Hyperbolic geometry, Möbius transformations, and geometric optimization.
D. Eppstein.
Invited talk at MSRI Introductory Worksh. Discrete & Computational Geometry, Berkeley, CA, 2003.Describes extensions of computational geometry algorithms to hyperbolic geometry, including an output-sensitive 3d Delaunay triangulation algorithm of Boissonat et al. and my own research on optimal Möbius transformation.
Minimum dilation stars.
D. Eppstein and K. Wortman.
arXiv:cs.CG/0412025.
21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 321–326.
Comp. Geom. Theory & Applications 37 (1): 27–37, 2007.We show how to test the dilation of a star, embedded in a Euclidean space of bounded dimension, in time O(n log n), and how to find the star center that has the minimum dilation for a given set of leaf points in randomized expected time O(n log n). For two-dimensional points, we can find the minimum dilation center, constrained to be one of the input points, in time O(n 2α(n) log2n). The unconstrained center placement algorithm involves quasiconvex programming, and is used as a subroutine in the constrained center placement algorithm.