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 follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(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 ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.

Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

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...

. Although it is NP-complete to decide whether a linear programming problemis degenerate, the ffl-perturbation method can be used to reduce in polynomial time any linear programming problemwith rational coefficients to a nondegenerate problem. The perturbed problem has the same status as the given one in terms of feasibility and unboundedness, and optimal bases of the perturbed problem are optimal in the given problem. Keywords: linear programming, polynomial-time, perturbation, degeneracy. OR/MS Index: 650 1. Introduction Degenerate problems cause some inconvenience in the practice as well as in the theory of linear programming. However, in this note we are interested only in the theoretical side. Many methods are known for avoiding the evils caused by degeneracy in the context of the simplex method (see, for example, Murty [3]). When a new algorithm is proposed, the analysis is often complicated by the need to address degeneracy, and results are sometimes proved under a nondegenerac...

This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with complementarity problems. The most important algorithms are presented along with a discussion of when they can be used eectively. We also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems. Finally, we provide a brief summary of current research trends. Key words: complementarity problems,variational inequalities, matrix classes 1 Introduction The distinguishing feature of a complementarity problem is the set of complementarity conditions. Each of these conditions requires that the product of two or more nonnegative quantities should be zero. (Here, each quantity is either a decision variable, or a function of the decisi...

We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization and maximization, and a randomized algorithm for optimizing arbitrary objectives.

The classes of P -, P 0 -, R 0 -, semimonotone, strictly semimonotone, column sufficient, and nondegenerate matrices play important roles in studying solution properties of equations and complementarity problems and convergence /complexity analysis of methods for solving these problems. It is known that the problem of deciding whether a square matrix with integer/rational entries is a P - (or nondegenerate) matrix is co-NP-complete. We show, through a unified analysis, that analogous decision problems for the other matrix classes are also co-NP-complete. Key words. P -, P 0 -, R 0 -, semimonotone, strictly semimonotone, column sufficient, nondegenerate matrices, complementarity problems, 1-norm maximzation, NP- completeness. This research is supported by National Science Foundation Grant CCR-9731273. y Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A. Email: tseng@math.washington.edu. 1 Introduction There are a number of matrix classes, in addition t...

In this chapter we present the basic mathematical results on the LCP. Many of these results are used in later chapters to develop algorithms to solve LCPs, and to study the computational complexity of these algorithms. Here, unless stated otherwise, I denotes the unit matrix of order n. M is a given square matrix of order n. In tabular form the LCP (q � M) is w z q I;M q w> 0 � z> 0 � w T z =0 (3:1)

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We show that it is NP-Complete to decide whether a bimatrix game is degenerate and it is Co-NP-Complete to decide whether a bimatrix game is nondegenerate. 1