The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing (i) all triangulations of a set of n points in the plane, (ii) all cells in a hyperplane arrangement in R d , (iii) all spanning trees of a graph, (iv) all Euclidean (non-crossing) trees spanning a set of n points in the plane, (v) all connected induced subgraphs of a graph, and (vi) all topological orderings of an acyclic graph. Finally we propose a new algorithm for the 0-1 integer programming problem which can be considered as an alternative to the branch-and-bound algorithm. 1 Introduction The listing of all objects that satisfy a specified property is a fundamental problem in combinatorics, computational geometr...

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

Abstract. Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tu-berlin.de/diskregeom/polymake/.

A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.

. The double description method is a simple and useful algorithm for enumerating all extreme rays of a general polyhedral cone in IR d , despite the fact that we can hardly state any interesting theorems on its time and space complexities. In this paper, we reinvestigate this method, introduce some new ideas for efficient implementations, and show some empirical results indicating its practicality in solving highly degenerate problems. 1 Introduction A pair (A; R) of real matrices A and R is said to be a double description pair or simply a DD pair if the relationship Ax 0 if and only if x = R for some 0 holds. Clearly, for a pair (A; R) to be a DD pair, it is necessary that the column size of A is equal to the row size of R, say d. The term "double description" was introduced by Motzkin et al. [MRTT53], and it is quite natural in the sense that such a pair contains two different descriptions of the same object. Namely, the set P (A) represented by A as P (A) = fx 2 IR d : Ax...

This paper is a survey and exposition of linear methods for finding Nash equilibria. Above all, these apply to games with two players. In an equilibrium of a twoperson game, the mixed strategy probabilities of one player equalize the expected payoffs for the pure strategies used by the other player. This defines an optimization problem with linear constraints. We do not consider nonlinear methods like simplicial subdivision for approximating fixed points, or systems of inequalities for higher-degree polynomials as they arise for noncooperative games with more than two players. These are surveyed in McKelvey and McLennan (1996)

This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for d-dimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all integer pivoting. It can also be used to compute the volume of the convex hull of a set of points. For a polyhedron with m inequalities in d variables and known extreme point, it finds all bases in time O(md²) per basis. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex and the complexity reduces to O(md) per vertex). Computational experience is included in the paper, including a comparison with an earlier implementation.

This paper describes three techniques for reachability analysis for systems modeled by ordinary differential equations (ODEs). First, linear models with regions modeled by convex polyhedra are considered, and an exact algorithm is presented. Next, non-convex polyhedra are considered, and techniques are presented for representing a polyhedron by its projection onto two-dimensional subspaces. This approach yields a compact representation, and allows efficient algorithms from computational geometry to be employed. Within this context, an approximation technique for reducing non-linear ODE models to linear nonhomogeneous models is presented. This

Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal--dual ...