Negotiation is a crucial part of commercial activities in physical as well as in electronic markets (see section 1.3). Current human-based negotiation is relatively slow, does not always uncover the best solution, and is, furthermore, constrained by issues of culture, ego, and pride (Beam and Segev, 1997). Experiments and field studies demonstrate that even in simple negotiations people often reach suboptimal agreements, ther eby “leaving money on the table” (Camerer, 1990). The end result is that the negotiators are often not able to reach agreements that would make each party better off.

The fact that negotiators fail to find better agreements highlights that negotiation is a search process. What makes negotiation different from the usual optimization search is that each side has private information, but neither typically knows the other's utility function. Furthermore, both sides often have an incentive to misrepresent their preferences (Oliver, 1997). Finding an optimal agreement in this environment is extremely challenging. Both sides are in competition but must jointly search for possible agreements. Although researchers in economics, game theory and behavioral sciences have investigated negotiation processes for a long time, a solid and comprehensive theoretical framework is still lacking. A basic principle of microeconomics and negotiation sciences is that there is not a single, “best” negotiation protocol for all possible negotiation situations.

This chapter collects open problems that in one way or the other relate to the material discussed in this book. They represent the complement of the material, in the sense that they attempt to describe what we do not know. We should keep in mind that it is most likely the case that only a tiny fraction of the knowable is known. Hence, there is a vast variety of questions that can be asked but not yet answered. The author of this book exercised subjective taste and judgment to collect a small subset of such questions, in the hope that they can give a glimpse of what is conceivable. Most of the problems are elementary in nature and have been stated elsewhere in the literature.

Two of the twenty-three problems have been solved since this book first appeared in 2001. These are P.8 Union of disks and P.9 Intersection of disks, both solved in [1]. Since the described approaches to the two problems are different from the eventual solution and perhaps useful to understand generalized versions of the problem, we decided to leave Sections P.8 and P.9 unchanged.

Empty convex hexagons

Let S be a set of n points in ℝ 2 and assume no three points are collinear. A convex k-gon is a subset of k points in convex position.

This chapter describes an algorithm for simplifying a given triangulated surface. We assume this surface represents a shape in three-dimensional space, and the goal is to represent approximately the same shape with fewer triangles. The particular algorithm combines topological and numerical computations and provides an opportunity to discuss combinatorial topology concepts in an applied situation. Section 4.1 describes the algorithm, which greedily contracts edges until the number of triangles that remain is as small as desired. Section 4.2 studies topological implications and characterizes edge contractions that preserve the topological type of the surface. Section 4.3 interprets the algorithm as constructing a simplicial map and establishes connections between the original and the simplified surfaces. Section 4.4 explains the numerical component of the algorithm used to prioritize edges for contraction.

Edge contraction algorithm

A triangulated surface is simplified by reducing the number of vertices. This section presents an algorithm that simplifies by repeated edge contraction. We discuss the operation, describe the algorithm, and introduce the error measure that controls which edges are contracted and in what sequence.

Edge contraction

Let K be a 2-complex, and assume for the moment that |K| is a 2-manifold. The contraction of an edge ab ∈ K removes ab together with the two triangles abx, aby, and it mends the hole by gluing xa to xb and ya to yb, as shown in Figure 4.1. Vertices a and b are glued to form a new vertex c.

The three sections in this chapter apply what we learned in Chapter 1 to the construction of triangle meshes in the plane. In mesh generation, the vertices are no longer part of the input but have to be placed by the algorithm itself. A typical instance of the meshing problem is given as a region, and the algorithm is expected to decompose that region into cells or elements. This chapter focuses on constructing meshes with triangle elements, and it pays attention to quality criteria, such as angle size and length variation. Section 2.1 shows how Delaunay triangulations can be adapted to constraints given as line segments that are required to be part of the mesh. Section 2.2 and 2.3 describe and analyze the Delaunay refinement method that adds new vertices at circumcenters of already existing Delaunay triangles.

Constrained triangulations

This section studies triangulations in the plane constrained by edges specified as part of the input. We show that there is a unique constrained triangulation that is closest, in some sense, to the (unconstrained) Delaunay triangulation.

Constraining line segments

The preceding sections constructed triangulations for a given set of points. The input now consists of a finite set of points, S ⊆ ℝ2, together with a finite set of line segments, L, each connecting two points in S. We require that any two line segments are either disjoint or meet at most in a common endpoint.

The primary purpose of this chapter is the introduction of standard topological language to streamline our discussions on triangulations and meshes. We will spend most of the effort to develop a better understanding of space, how it is connected, and how we can decompose it. The secondary purpose is the construction of a bridge between continuous and discrete concepts in geometry. The idea of a continuous and possibly even differential world is close to our intuitive understanding of physical phenomena, while the discrete setting is natural for computation. Section 3.1 introduces simplicial complexes as a fundamental discrete representation of continuous space. Section 3.2 talks about refining complexes by decomposing simplices into smaller pieces. Section 3.3 describes the topological notion of space and the important special case of manifolds. Section 3.4 discusses the Euler characteristic of a triangulated space.

Simplicial complexes

We use simplicial complexes as the fundamental tool to model geometric shapes and spaces. They generalize and formalize the somewhat loose geometric notions of a triangulation. Because of their combinatorial nature, simplicial complexes are perfect data structures for geometric modeling algorithms.

Simplices

A finite collection of points is affinely independent if no affine space of dimension i contains more than i + 1 of the points, and this is true for every i. A k-simplex is the convex hull of a collection of k + 1 affinely independent points, σ = conv S.