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23
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
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Cited by 1213 (77 self)
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the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
Social Potential Fields: A Distributed Behavioral Control for Autonomous Robots
, 1999
"... A Very Large Scale Robotic (VLSR) system may consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and ..."
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Cited by 183 (1 self)
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A Very Large Scale Robotic (VLSR) system may consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and military applications of VLSR systems in tasks such as assembling, transporting, hazardous inspection, patrolling, guarding and attacking. In this paper, we propose a new approach for distributed autonomous control of VLSR systems. We define simple artificial force laws between pairs of robots or robot groups. The force laws are inversepower force laws, incorporating both attraction and repulsion. The force laws can be distinct and to some degree they reflect the 'social relations' among robots. Therefore we call our method social potential fields. An individual robot's motion is controlled by the resultant artificial force imposed by other robots and other components of the system. The approach is distributed in that the force calculations and motion control can be done in an asynchronous and distributed manner. We also extend the social potential fields model to use spring laws as force laws. This paper presents the first and a preliminary study on applying potential fields to distributed autonomous multirobot control. We describe the generic framework of our social potential fields method. We show with computer simulations that the method can yield interesting and useful behaviors among robots, and we give examples of possible industrial and military applications. We also identify theoretical problems for future studies. 1999 Published by Elsevier Science B.V. All rights reserved.
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 29 (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 ...
Approximation Algorithms for CurvatureConstrained Shortest Paths
, 1996
"... Let B be a point robot in the plane, whose path is constrained to have curvature of at most 1, and let\Omega be a set of polygonal obstacles with n vertices. We study the collisionfree, optimal pathplanning problem for B. Given a parameter ", we present an O((n 2 =" 2 ) log n)time a ..."
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Cited by 24 (4 self)
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Let B be a point robot in the plane, whose path is constrained to have curvature of at most 1, and let\Omega be a set of polygonal obstacles with n vertices. We study the collisionfree, optimal pathplanning problem for B. Given a parameter ", we present an O((n 2 =" 2 ) log n)time algorithm for computing a collisionfree, curvatureconstrained path between two given positions, whose length is at most (1 + ") times the length of an optimal robust path (a path is robust if it remains collisionfree even if certain positions on the path are perturbed). Our algorithm thus runs significantly faster than the previously best known algorithm by Jacobs and Canny whose running time is O(( n+L " ) 2 + n 2 ( n+L " ) log n), where L is the total edge length of the obstacles. More importantly, the running time of our algorithm does not depend on the size of obstacles. The path returned by this algorithm is not necessarily robust. We present an O((n=") 2:5 log n) time algorithm that...
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 22 (10 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 13 (5 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety
 J. Complexity
"... We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then ..."
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Cited by 8 (4 self)
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We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then we prove PSPACEhardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer. Key words: connected components, Betti numbers, PSPACE, lower bounds 1
Samplingbased optimal motion planning for nonholonomic dynamical systems,” in
 Proc. IEEE Conf. on Robotics and Automation,
, 2013
"... AbstractSamplingbased motion planning algorithms, such as the Probabilistic RoadMap (PRM) and the Rapidlyexploring Random Tree (RRT), have received a large and growing amount of attention during the past decade. Most recently, samplingbased algorithms, such as the PRM * and RRT * , that guarante ..."
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Cited by 7 (2 self)
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AbstractSamplingbased motion planning algorithms, such as the Probabilistic RoadMap (PRM) and the Rapidlyexploring Random Tree (RRT), have received a large and growing amount of attention during the past decade. Most recently, samplingbased algorithms, such as the PRM * and RRT * , that guarantee asymptotic optimality, i.e., almostsure convergence towards optimal solutions, have been proposed. Despite the experimental success of asymptoticallyoptimal samplingbased algorithms, their extensions to handle complex nonholonomic dynamical systems remains largely an open problem. In this paper, with the help of results from differential geometry, we extend the RRT * algorithm to handle a large class of nonholonomic dynamical systems. We demonstrate the performance of the algorithm in computational experiments involving the Dubins' car dynamics.
On the Complexity of Counting Irreducible Components and Computing Betti Numbers of Algebraic Varieties
, 2007
"... This thesis is a continuation of the study of counting problems in algebraic geometry within an algebraic framework of computation started by Bürgisser, Cucker, and Lotz in a series of papers [BC03, BC06, BCL05]. In its first part we give a uniform method for the two problems #CCC and #ICC of count ..."
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Cited by 3 (1 self)
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This thesis is a continuation of the study of counting problems in algebraic geometry within an algebraic framework of computation started by Bürgisser, Cucker, and Lotz in a series of papers [BC03, BC06, BCL05]. In its first part we give a uniform method for the two problems #CCC and #ICC of counting the connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szánto ́ [Szá97] computing a variant of characteristic sets. The second part contains lower bounds in terms of hardness results for topological problems dealing with complex algebraic varieties. In particular, we show that the problem of deciding connectedness of a complex affine or projective va