3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting—preprocess a set of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure—preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching— preprocess a set C of n convex fat polygons, so that the k objects intersecting a “not-too-large” query polygon can be reported efficiently, and (iv) bounded-size segment shooting—preprocess a set C as in (iii), so that the first object (if exists) hit by a “not-too-long” oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(λ_s(n)log³n), where s is the maximum number of intersections between the boundaries of the (xy-projections) of any pair of objects, and λ_s(n) is the maximum length of (n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(λ_s(n)log²n). The query time in the first problem is O(log⁴n), the query time in the second and third problems is O(log³n + klog²n), and the query time in the fourth problem is O(log³n).

We also present a simple algorithm for computing a depth order for a set as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for , if exists, is a linear order of , such that, if K₁, K₂ and K₁ lies vertically above K₂, then K₁ precedes K₂.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm.     

A fast approximation algorithm for solving the complete set packing problem

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.     

The ?-positive center figure

Based on Gage's notion of a “positive center” of a planar convex set, an ε-positive center figure is defined, constructed, and illustrated through example. The existence of an ε-positive center point, and the convexity of the ε-positive center figure, is conjectured.     

H least-squares method for the velocity–pressure–stress formulation of Stokes equations

This paper studies the discrete H⁻¹-norm least-squares method for the incompressible Stokes equations based on the velocity–pressure–stress formulation by the least-squares functional defined as the sum of L²-norms and H⁻¹-norm of the residual equations. Some computational experiments by multigrid method and preconditioning conjugate gradient method (PCGM) on this method are shown by taking efficient and β in the discrete solution operator T_h=h²I+βB_h corresponding to the minus one norm. We also propose a new method and compare it with PCGM and multigrid method through the analysis of numerical experiments depending on the choice of β.     

Analytic efficient solution set for multi-criteria quadratic programs

We present active set methods to evaluate the exact analytic efficient solution set for multi-criteria convex quadratic programming problems (MCQP) subject to linear constraints. The idea is based on the observations that a strictly convex programming problem admits a unique solution, and that the efficient solution set for a multi-criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized. The case of bi-criteria quadratic programming (BCQP) is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives. In particular we note that the efficient solution set of a BCQP problem is a curve on the surface of a polytope. The extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation. Two numerical examples are given to illustrate the proposed methods.     

Improved Bounds on the Union Complexity of Fat Objects

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ _s+2(n)log ² n). This improves the best known bound, and extends it to a more general class of objects. The research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

Offset-polygon annulus placement problems

An offset-polygon annulus region is defined in terms of a polygon P and a distance δ > 0 (offset of P). In this paper we solve several containment problems for polygon annulus regions with respect to an input point set. Optimization criteria include both maximizing the number of points contained in a fixed size annulus and minimizing the size of the annulus needed to contain all points. We address the following variants of the problem: placement of an annulus of a convex polygon as well as of a simple polygon; placement by translation only, or by translation and rotation; off-line and on-line versions of the corresponding decision problems; and decision as well as optimization versions of the problems. We present efficient algorithms in each case.     

On convex segments in a triangulation

A segment (=1-cell) of a planar triangulation σ is convex if it is common to two triangles (2-cells) whose union is a convex set. We determine the maximal number of convex segments of a triangulation over all triangulations σ having n boundary vertices and m inner vertices (n3,m0).     

Karp complexity and classes with the independence property

A class K of structures is controlled if for all cardinals λ, the relation of L_∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see Arch. Math. Logic 40 (2001) 69–88).     

Non-crossing matchings of points with geometric objects

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point–object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point–object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.     

Algorithms for generalized halfspace range searching and other intersection searching problems

In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in , for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.     

Method of reduction in convex programming

We present an algorithm which solves a convex program with faithfully convex (not necessarily differentiable) constraints. While finding a feasible starting point, the algorithm reduces the program to an equivalent program for which Slater's condition is satisfied. Included are algorithms for calculating various objects which have recently appeared in the literature. Stability of the algorithm is discussed.     

A geometric inequality on mixed volumes

We develop some geometric inequality for a kind of generalized convex set. The integral of (n – 2)-th mean curvature of the generalized convex set, the mixed volume of the convex hull of the set, and a reference convex set are involved in the inequality.Partially supported by grants from Kosef and BSRI-95-1419.     

On fat partitioning, fat covering and the union size of polygons

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n²) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O(n log log n), which is a generalization of a similar result for fat triangles (Matou ek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.     

Decompositions and boundary coverings of non-convex fat polyhedra

We show that any locally-fat (or (α,β)-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require Ω(n²) pieces in any convex decomposition. Furthermore, we show that if we want the tetrahedra in the decomposition to be fat themselves, then their number cannot be bounded as a function of n in the worst case. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.     

A Helly-type theorem for higher-dimensional transversals

We prove that a collection of compact convex sets of bounded diameters in that is unbounded in k independent directions has a k-flat transversal for k<d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1.     

Search for Efficient Solutions of Multi-Criterion Problems by Target-Level Method

The target-level method is considered for solving continuous multi-criterion maximization problems. In the first step, the decision-maker specifies a target-level point (the desired criterion values); then in the set of vector evaluations we seek points that are closest to the target point in the Chebyshev metric. The vector evaluations obtained in this way are in general weakly efficient. To identify the efficient evaluations, the second step maximizes the sum of the criteria on the set generated in step 1. We prove the relationship between the evaluations and decisions obtained by the proposed procedure, on the one hand, and the efficient (weakly efficient) evaluations and decisions, on the other hand. If the Edgeworth–Pareto hull of the set of vector evaluations is convex, the set of efficient vector evaluations can be approximated by the proposed method.     

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x₀,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence (i_j)_{0 j n} of ordinals such that i₀ = β, i_n = and x_ij+1 is adjacent with x_ij and with all neighbors of x_ij in the subgraph ofG induced by {x_y: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.