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39
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
Composition of Local Potential Functions for Global Robot Control and Navigation
, 2003
"... This paper develops a method of composing simple control policies, applicable over a limited region in a dynamical system's free space, such that the resulting composition completely solves the navigation and control problem for the given system operating in a constrained environment. The resulting ..."
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Cited by 48 (5 self)
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This paper develops a method of composing simple control policies, applicable over a limited region in a dynamical system's free space, such that the resulting composition completely solves the navigation and control problem for the given system operating in a constrained environment. The resulting control policy deployment induces a global control policy that brings the system to the goal, provided that there is a single connected component of the free space containing both the start and goal configurations. In this paper, control policies for both kinematic and simple dynamical systems are developed. This work assumes that the initial velocities are somewhat aligned with the desired velocity vector field. We conclude by offering an outline of an approach for accommodating arbitrary dynamical constraints and initial conditions.
Randomized PursuitEvasion in a Polygonal Environment
, 2004
"... This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The ..."
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Cited by 48 (7 self)
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This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The evader can be arbitrarily faster than the pursuer and it may know the position of the pursuer at all times but it does not have prior knowledge of the random decisions made by the pursuer. Second, using the randomized algorithm together with the solution to a problem called the "lion and man problem" [2] as subroutines, we present a strategy for two pursuers (one of which is at least as fast as the evader) to quickly capture an evader in a simplyconnected polygonal environment. We show how this strategy can be extended to obtain a strategy for (i) a polygonal room with a door, (ii) two pursuers who have only lineofsight communication, and (iii) a single pursuer (at the expense of increased capture time).
Polygon Decomposition for Efficient Construction of Minkowski Sums
, 2000
"... Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various ..."
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Cited by 40 (7 self)
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Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various wellknown decompositions as well as with several new decomposition schemes. We report on our experiments with various decompositions and different input polygons. Among our findings are that in general: (i) triangulations are too costly (ii) what constitutes a good decomposition for one of the input polygons depends on the other input polygon  consequently, we develop a procedure for simultaneously decomposing the two polygons such that a "mixed" objective function is minimized, (iii) there are optimal decomposition algorithms that significantly expedite the Minkowskisum computation, but the decomposition itself is expensive to compute  in such cases simple heuristics that approximate the optimal decomposition perform very well.
AN O(n log log n)TIME ALGORITHM FOR TRIANGULATING A SIMPLE POLYGON
, 1988
"... Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that tria ..."
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Cited by 37 (4 self)
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Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.
Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami
 Computational Geometry: Theory and Applications
, 1999
"... We show a remarkable fact about folding paper: From a single rectangular sheet of paper, one can fold it into a flat origami that takes the (scaled) shape of any connected polygonal region, even if it has holes. This resolves a longstanding open problem in origami design. Our proof is constructive, ..."
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Cited by 28 (12 self)
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We show a remarkable fact about folding paper: From a single rectangular sheet of paper, one can fold it into a flat origami that takes the (scaled) shape of any connected polygonal region, even if it has holes. This resolves a longstanding open problem in origami design. Our proof is constructive, utilizing tools of computational geometry, resulting in efficient algorithms for achieving the target silhouette. We show further that if the paper has a different color on each side, we can form any connected polygonal pattern of two colors. Our results apply also to polyhedral surfaces, showing that any polyhedron can be “wrapped ” by folding a strip of paper around it. We give three methods for solving these problems: the first uses a thin strip whose area is arbitrarily close to optimal; the second allows wider strips to be used; and the third varies the strip width to optimize the number or length of visible “seams ” subject to some restrictions. Key words: paper folding, origami design, polyhedra, polyhedral surfaces, Hamiltonian triangulation, straight skeleton, convex decomposition
Compaction Algorithms for NonConvex Polygons and Their Applications
, 1994
"... Given a twodimensional, nonoverlapping layout of convex and nonconvex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial twodimensional packing applications, compaction can improve the material utilization of already ti ..."
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Cited by 27 (2 self)
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Given a twodimensional, nonoverlapping layout of convex and nonconvex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial twodimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of nonconvex polygons are not previously known. This dissertation offers the first systematic study of compaction of nonconvex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACEhard. The major contribution of this dissertation is a positionbased optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first ...
On the Time Bound for Convex Decomposition of Simple Polygons
 SCHOOL OF COMPUTER SCIENCE, MCGILL UNIVERSITY
, 1998
"... We show that a decomposition of a simple polygon having n vertices, r of which are reflex, into a minimum number of convex regions without the addition of Steiner vertices can be computed in O(n + r minfr ; ng) time and space. A Java demo is available at http://www.cs.ubc.ca/ spider/snoeyin ..."
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Cited by 24 (1 self)
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We show that a decomposition of a simple polygon having n vertices, r of which are reflex, into a minimum number of convex regions without the addition of Steiner vertices can be computed in O(n + r minfr ; ng) time and space. A Java demo is available at http://www.cs.ubc.ca/ spider/snoeyink/demos/convdecomp
Approximate convex decomposition of polygons
 In Proc. 20th Annual ACM Symp. Computat. Geom. (SoCG
, 2004
"... We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller ..."
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Cited by 22 (3 self)
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We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing (resolving) the most significant nonconvex feature (notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex ’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O(nr) time. In contrast, exact convex decomposition is NPhard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our webpages at: