Small grid embeddings of 3-polytopes

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 $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that 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

Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

Resolving Loads with Positive Interior Stresses

We consider the pair (pi, fi) as a force with two-dimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a non-crossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, such that for every pi the sum of the weighted edges (seen as vectors) around pi yields −fi. As additional constraint we restrict ourselves to weights that are non-negative on every edge that is not on the convex hull of the point set. We show that (under a generic assumption) for any reasonable set of forces there is exactly one pointed pseudo-triangulation that fulfils the desired properties. Our results will be obtained by linear programming duality over the PPT-polytope. For the case where the forces appear only at convex hull vertices we show that the pseudo-triangulation that resolves the load can be computed as weighted Delaunay triangulation. Our observations lead to a new characterization of pointed pseudo-triangulations, structures that have been proven to be extremely useful in the design and analysis of efficient geometric algorithms. As an application, we discuss how to compute the maximal locally convex function for a polygon whose corners lie on its convex hull

A Global Parametric Programming Optimisation Strategy for Multilevel Problems

In this paper, we outline the foundations of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems based on our recent developments in multiparametric programming theory. The core idea is to recast each optimisation subproblem in the multilevel hierarchy as a multiparametric programming problem and then transform the multilevel problem into a single-level optimisation problem. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments