It is NP-complete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NP-complete to recognize whether two sets of points in the plane can be separated with k lines. For every fixed k in any fixed dimension, it takes polynomial time to recognize whether two sets of points can be separated with k hyperplanes.

Predicate abstraction has emerged to be a powerful technique for extracting finite-state models from infinite-state systems, and has been recently shown to enhance the effectiveness of the reachability computation techniques for hybrid systems. Given a hybrid system with linear dynamics and a set of linear predicates, the verifier performs an on-the-fly search of the finite discrete quotient whose states correspond to the truth assignments to the input predicates. The success of this approach crucially depends on the choice of the predicates used for abstraction. In this paper, we focus on identifying these predicates automatically by analyzing spurious counter-examples generated by the search in the abstract state-space. We present the basic techniques for discovering new predicates that will rule out closely related spurious counter-examples, optimizations of these techniques, implementation of these in the verification tool, and case studies demonstrating the promise of the approach.

Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NP-complete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwise-disjoint boundaries such that Pi Ri andP jRij m. In three dimensions, the problem is NP-complete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present e cient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedron from a nonconvex polyhedron with a polyhedral surface whose facet-complexity is O(log n) times the optimal, where n = jPj+jQj is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O(n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d 4, the facet-complexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by non-polyhedral surfaces, such as spherical patches.

We show that several well-known optimization problems involving 3-dimensional convex polyhedra and decision trees are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computer-aided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3-connected. Unfortunately, algorithmic problems dealing with 3-dimensional convex polyhedra ...

In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface that separates two sets. In this paper, we provide approximation algorithms for several geometric separation problems, including: • Given a set of triangles T and a set S of points that lie within the union of the triangles, find a minimum-cardinality set, T ′ , of pairwise-disjoint triangles, each contained within some triangle of T, that cover the point set S. • Given finite sets of “red ” and “blue ” points in the plane, determine a simple polygon of fewest edges that separates the red points from the blue points. More generally, given finite sets of points of many color classes, determine a planar “separating ” subdivision of minimum combinatorial complexity, which has the property that each face of the subdivision contains points of at most one color class; • Given two polyhedral terrains, P and Q, over a common support set (e.g., the unit square), with P lying above Q, compute a nested polyhedral terrain R that lies between P and Q such that R has a minimum number of facets. Exact solution of the above problems in polynomial time is highly unlikely: The decision versions of all three problems are known to be NP-hard. We provide polynomial-time algorithms that are guaranteed to produce an answer within a logarithmic factor (O(log n), where n is the complexity of the input problem instance) of optimal. (The error factor is constant in the orthogonal case — coverage by disjoint aligned rectangles, or separation of orthohedral terrains.) We also discuss extensions to higher dimensions. 1

A circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An \Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect\Omega\Gamma n 2 ) times, our algorithm can be adapted to work in O(nff(n) log n) time and O(nff(n)) space, where ff(n) represents the extremely slowly growing inverse of the Ackermann function.

In this paper we study some problems on the separability of two disjoint point sets in the plane by means of several strips, wedges and sectors. An efficient algorithm for minimizing the number of missclassified points, when only one strip is used, and a lower bound for the fixed-slopes double wedge problem are also given.

In this paper we study the separability of two disjoint sets of objects in the plane according to two criteria: wedge separability and strip separability. Wegivealgorithms for computing all the separating wedges and strips, the wedges with the maximum and minimum angle, and the narrowest and the widest strip. The objects we consider are points, segments, polygons and circles. As applications, we improve the computation of all the largest circles separating two sets of line segments by a log n factor, and we generalize the algorithm for computing the minimum polygonal separator of two sets of points to two sets of line segments with the same running time. Key words: Red-blue separability, wedges, strips, circular and polygonal separability. 1

In this paper we obtain lower bounds in the algebraic computation tree model for deciding the separability of two disjoint point sets. More precisely, we show Ω(n log n) time lower bounds for the separability by means of strips, wedges, wedges with apices on a given line, fixed-slopes double wedges, and triangles (solving an open problem from Edelsbrunner and Preparata), which match the complexity of the existing algorithms, and therefore prove their optimality. 1

First, and foremost, I want to thank my advisor Professor Rajeev Alur. His knowl-edge and constant guidance have helped me a long way towards completing this thesis. I would also like to thank Professor Insup Lee for chairing my thesis com-mittee, and Professors Vijay Kumar, George Pappas, and Bruce Krogh from the Carnegie-Mellon University for accepting to be members on my thesis committee. Many thanks go out to Professor Oleg Sokolsky as well. In addition, I would like to thank the whole CIS department for making Penn such a fruitful experience to me. Special thanks go out to Mike Felker who was always helpful. During my time at Penn, I have collaborated with many researchers from the CIS department, as well as other departments of Penn, but also with members of other research organizations. Most importantly, I would like to thank Thao Dang, without whom most of this work would not have been implementable, and who also became a very close friend of mine in the process. Additionally, I would like to thank Eric Aaron, Calin Belta, Ansgar Fehnker, and Jesung Kim for various contributions to my research that is presented in this thesis. I also want to thank Maria Adamou, Dimos