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47
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.
Coordinated navigation of multiple independent diskshaped robots
 EECS Dept., Univ. Michigan, Ann Arbor, MI, Tech. Rep
, 2004
"... This paper addresses the coordinated navigation of multiple independently actuated diskshaped robots all placed within the same diskshaped workspace. We encode complete information about the goal, obstacles and workspace boundary using an artificial potential function over the cross product space ..."
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Cited by 11 (0 self)
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This paper addresses the coordinated navigation of multiple independently actuated diskshaped robots all placed within the same diskshaped workspace. We encode complete information about the goal, obstacles and workspace boundary using an artificial potential function over the cross product space of the robots ’ simultaneous configurations. The closedloop dynamics governing the motion of each robot take the form of the approriate projection of the gradient of this function. We show, with some reasonable restrictions on the allowable goal positions, that this function is an essential navigation function a special type of artificial potential function that is ensured of connecting the kinematic planning with the dynamic execution in a correct manner. Hence, each robot is guaranteed of collisionfree navigation to its destination from almost all initial free placements.
Topological Robotics: Motion Planning
, 2005
"... In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any pathconnected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X. ” We examine how the number TC( ..."
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Cited by 9 (1 self)
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In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any pathconnected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X. ” We examine how the number TC(X) determines the structure of motion planning algorithms, both deterministic and random. We compute the invariant TC(X) in many interesting examples. In the case of the real projective space �P n (where n � 1, 3, 7) the number TC(�P n) − 1 equals the minimal dimension of the Euclidean space into which �P n can be immersed.
On lines avoiding unit balls in three dimensions
, 2004
"... Let B be a set of n unit balls in R 3. We show that the combinatorial complexity of the space of lines in R 3 that avoid all the balls of B is O(n ..."
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Cited by 8 (0 self)
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Let B be a set of n unit balls in R 3. We show that the combinatorial complexity of the space of lines in R 3 that avoid all the balls of B is O(n
Semilinear motion planning in REDLOG
 AAECC
, 1999
"... We study a new type of motion planning problem in dimension 2 and 3 via linear and quadratic quantifier elimination. The object to be moved and the free space are both semilinear sets with no convexity assumptions. The admissible motions are finite continuous sequences of translations along prescrib ..."
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Cited by 7 (2 self)
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We study a new type of motion planning problem in dimension 2 and 3 via linear and quadratic quantifier elimination. The object to be moved and the free space are both semilinear sets with no convexity assumptions. The admissible motions are finite continuous sequences of translations along prescribed directions. When the number of translations is bounded in advance, then the corresponding path finding problem can be modelled and solved as a linear quantifier elimination problem. Moreover the problem to find a shortest or almost shortest admissible path can be modelled as a special quadratic quantifier elimination problem. We give upper complexity bounds on these problems, report experimental results using the elimination facilities of the redlog package of reduce, and indicate a possible application.
Approximation Algorithms for Aligning Points
 IN SOCG'03
, 2003
"... We study the problem of aligning as many points as possible horizontally, vertically, or diagonally, when each point is allowed to be placed anywhere in its own, given region. Different shapes of placement regions and different sets of alignment orientations are considered. More generally, we assume ..."
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Cited by 7 (2 self)
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We study the problem of aligning as many points as possible horizontally, vertically, or diagonally, when each point is allowed to be placed anywhere in its own, given region. Different shapes of placement regions and different sets of alignment orientations are considered. More generally, we assume that a graph is given on the points, and only the alignments of points that are connected in the graph count. We show that for planar graphs the problem is NPhard, and we provide an inapproximability result for general graphs. For the case of trees and planar graphs, we give approximation algorithms whose performance depends on the shape of the given regions and the set of orientations. When the orientations are the ones given by the axes and the regions are axisparallel rectangles, we obtain a polynomial time approximation scheme.
On the exact maximum complexity of Minkowski sums of convex polyhedra
 PROCEEDINGS OF 23RD ANNUAL ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY (SOCG
, 2007
"... We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given ..."
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Cited by 7 (2 self)
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We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1, m2,..., mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is � 1≤i<j≤k (2mi − 5)(2mj − 5) + � � k � 2 + 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several preconstructed convex polyhedra, the Minkowski sum of which is maximal, are available at http://www.cs.tau.ac.il / ~ efif/Mink. The polyhedra models and an interactive program that computes their Minkowski sums and visualizes them can be downloaded as well.
Planning Highquality Paths and Corridors Amidst Obstacles �
"... The motionplanning problem, involving the computation of a collisionfree path for a moving entity amidst obstacles, is a central problem in fields such robotics and game design. In this paper we study the problem of planning highquality paths. A highquality path should have some desirable proper ..."
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Cited by 6 (1 self)
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The motionplanning problem, involving the computation of a collisionfree path for a moving entity amidst obstacles, is a central problem in fields such robotics and game design. In this paper we study the problem of planning highquality paths. A highquality path should have some desirable properties: it should be short, avoiding long detours, and at the same time it should stay at a safe distance from the obstacles, namely it should have clearance. We suggest a quality measure for paths, which balances between the above criteria of minimizing the path length while maximizing its clearance. We analyze the properties of optimal paths according to our measure, and devise an approximation algorithm to compute nearoptimal paths amidst polygonal obstacles in the plane. We also apply our quality measure to corridors. Instead of planning a onedimensional motion path for a moving entity, it is often more convenient to let the entity move in a corridor, where the exact motion path is determined by a local planner. We show that planning an optimal corridor is equivalent to planning an optimal path with bounded clearance. KEY WORDS—Motion planning, bicriteria optimization, corridors
Smoothed Analysis of Probabilistic Roadmaps
"... The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples req ..."
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Cited by 6 (0 self)
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The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples required to produce an accurate probabilistic roadmap, and thus on the running time of the algorithm, in an environment of simplices. This sheds light on its widespread empirical success.
Topology preserving approximation of free configuration space
 In Proceedings of International Conference on Robotics and Automation
, 2006
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