Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

INTRODUCTION Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independently, and a two-dimensional or threedimensional environment V cluttered with obstacles. We assume that the shape and location of the obstacles and the shape of B are known to the planning system. Given an initial placement Z 1 and a nal placement Z 2 of B, we wish to determine whether there exists a collision-avoiding motion of B from Z 1 to Z 2 , and, if so, to plan such a motion. In this simpli ed and purely geometric setup, we ignore issues such as incomplete information, nonholonomic constraints, control issues related to inaccuracies in sensing and motion, nonstationary obstacles, optimality of the planned motion, and so on. Since the early 1980's, motion planning has been an intensiv

The motion planning problems for non-holonomic car-like robots have been extensively studied in the literature. The curvature-constrained shortest-path problem is to plan a path (from an initial configuration to a final configuration, where a configuration is defined by a location and an orientation) in the presence of obstacles, such that the path is a shortest among all paths with a prescribed curvature bound. The curvature-constrained shortest-path problem can also be seen as finding a shortest path for a point car-like robot moving forward at constant speed with a radius of curvature upper bounded by some constant. Previously, there is no known hardness result for the 2D curvature constrained shortest-path problem. This paper shows that the above problem in two dimensions is NP-hard, when the obstacles are polygons with a total of N vertices and the vertex positions are given within O(N²) bits of precision. Our reduction is computed by a family of polynomial-size ...

We introduce a technique for computing approximate solutions to optimization problems. If X is the set of feasible solutions, the standard goal of approximation algorithms is to compute x X that is an #-approximate solution in the following sense: (1 + #)d(x # ) where x # X is an optimal solution, d : X R#0 is the optimization function to be minimized, and # > 0 is an input parameter. Our approach is to first devise algorithms that compute pseudo #-approximate solutions satisfying the bound d(x # R ) + #R where R > 0 is a new input parameter. Here x # R denotes an optimal solution in the space XR of R-constrained feasible solutions. The parameterization provides a stratification of X in the sense that (1) XR XR # , for R < R # and (2) XR = X for R su#ciently large.

We accept this thesis as conforming

In this paper we define a class of mechanical systems consisting of rigid objects (defined by linear or quadratic surface patches) connected by frictional contact linkages between surfaces. (This class of mechanisms is similar to the Analytical Engine developed by Babbage in 1800s except that we assume frictional surfaces instead of toothed gears.) We prove that a universal Turing Machine (TM) can be simulated by a (universal) frictional mechanical system in this class consisting of a constant number of parts. Our universal frictional mechanical system has the property that it can reach a distinguished final configuration through a sequence of legal movements if and only if the universal TM accepts the input string encoded by its initial configuration. There are two implications from this result. First, the robotic mover's problem is undecidable when there are frictional linkages. Second, a mechanical computer can be constructed that has the computational power of any conventional electronic computer and yet has only a constant number of mechanical parts. Previous

We continue, and in a sense complete, our study the motion of a rod (line segment) in the plane in the presence of polygonal obstacles, under an optimality criterion based on minimizing the trace length of a fixed but arbitrary point (called the focus) onthe rod. In an earlier paper, we showed that this problem is NP-hard when the focus is in the relative interior of the rod. Our proof did not cover the case where the focus lies at one of the rod endpoints. Indeed, considerable evidence suggested that this special case might admit a polynomial time solution. In this paper we settle this open problem by proving, by means of a non-trivial adaptation of our earlier construction, that this remaining case is also NP-hard. 1