Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were:

this paper is (1) to give three new applications of these concepts to 2-dimensional visibility problems, and (2) to study realizability questions suggested by the pseudotriangle-pseudoline duality; see Figure 1. Our first application is related to the ray-shooting problem in the plane: preprocess a set of objects into a data structure such that the first object hit by a query ray can be computed efficiently. In section 3 we show that for a scene of n objects, where the objects are pairwise disjoint convex sets with m 'simple' arcs in total, one can obtain O(log m) query time using

ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some three-person game, and also to the set of totally mixed Nash equilibria of an N-person game in which each player has two pure strategies. From the Nash-Tognoli Theorem it follows that every compact differentiable manifold can be encoded as the set of totally mixed Nash equilibria of some game. Moreover, there exist isolated Nash equilibria of arbitrary topological degree. 1.

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...

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Let’s begin with a little “pep talk”, some (very) brief history, and some of the motivating examples of matroids. 1.1. Motivation. Why learn about or study matroids/oriented matroids in geometric, topological, algebraic combinatorics? Here are a few of my personal reasons. • They are general, so results about them are widely applicable. • They have relatively few axioms and standard constructions/techniques, so they focus one’s approach to solving a problem. • They give examples of well-behaved objects: polytopes, cell/simplicial complexes, rings. • They provide “duals ” for non-planar graphs! 1.2. Brief early history. (in no way comprehensive...) 1.2.1. Matroids. • H. Whitney (1932, 1935)- graphs, duality, and matroids as abstract linear independence.

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2 5.54n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.

Abstract. We show that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time. Our proof relies on a primal-dual characterization (by Joswig, Kaibel and Korner) and a linear program, with an exponential number of constraints, which can be used to construct the solution and can be solved in polynomial time. We show that this allows one to characterize the face lattice of the polytope, via a simple face recognition algorithm. In addition, we define the concept of a pseudo-polytopal-multigraph which may be of independent interest. 1

We show how to embed a 3-connected planar graph with n vertices as a 3-polytope with small integer coordinates. The coordinates are bounded by O(2 7.55n). The crucial part is the construction of a plane embedding which supports an equilibrium stress. We have to guarantee that the size of the coordinates and the stresses are small. This is achieved by applying Tutte’s spring embedding method carefully.

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2 5.54n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face. 1