An approximation scheme for planar graph TSP
We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.

title: Segmentation Problems

title: The complexity of recognizing polyhedral scenes
abstract: Given a drawing of straight lines on the plane, we wish to decide whether it is the projection of the visible part of a set of opaque polyhedra. Although there is an extensive literature and reports on empirically succesful algorithm: for this problem, there has been no definite result concerning its complexity. In this paper we show that, rather surprisingly, this problem is NP-complete. This is true even in the relatively simple case of trihedral scenes (no four planes share a point) without shadows or cracks. Despite this negative result, we present a fast algorithm for the important special case of orthohedral scenes (all planes are perpendicular to one of the three axes) with a fixed number of "possible" objects.

title: Optimum Grip of a Polygon
abstract: It has been shown by Baker, Fortune and Grosse that any two-dimensional polygonal object can be prehended stably with three fingers, so that its weight (along the third dimension) is balanced. Besides, in this paper we show that form closure of a polygon object can be achieved by four fingers (previous proofs were not complete). We formulate and solve the problem of finding the optimum stable grip or form closure of any given polygon. For stable grip it is most natural to minimize the forces needed to balance through friction the object''s weight along the third dimension. For form closure, we minimize the worst-case forces needed to balance any unit force acting on the center of gravity of the object. The mathematical techniques used in the two instances are an interesting mix of Optimization and Euclidean geometry. Our results lead to algorithms for the efficient computation of the optimum grip in each case.

title: On the complexity of equilibria
abstract: We prove complexity, approximability, and inapproximability results for the problem of finding an exchange equilibrium in markets with indivisible (integer) goods, most notably a polynomial-time algorithm that approximates the market equilibrium arbitrarily closely when the number of goods is bounded and the utilities are linear. We also show a communication complexity lower bound, implying that the ideal informational economy of a market with unique individual optima is unattainable in general.

title: Algorithms, Games, and the Internet
abstract: Over the past fifty years, researchers in Theoretical Computer Science have sought and achieved a productive foundational understanding of the von Neumann computer and its software, employing the mathematical tools of Logic and Combinatorics. The next half century appears now much more confusing (half- centuries tend to look like that in the beginning). What computational artifact will be the object of the next great modeling adventure of our field? And what mathematical tools will be handy in this endeavor?

title: Communication Complexity