On rectangular covering problems

Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension

On rectangular covering problems

Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension

Studies in Efficient Discrete Algorithms

This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity

QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems

Combinatorial optimization problems in self-assembly

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape. We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems

Skeletal representations of orthogonal shapes

In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version

Mapping, planning and exploration with Pose SLAM

This thesis reports research on mapping, path planning, and autonomous exploration. These are classical problems in robotics, typically studied independently, and here we link such problems by framing them within a common SLAM approach, adopting Pose SLAM as the basic state estimation machinery. The main contribution of this thesis is an approach that allows a mobile robot to plan a path using the map it builds with Pose SLAM and to select the appropriate actions to autonomously construct this map. Pose SLAM is the variant of SLAM where only the robot trajectory is estimated and where landmarks are only used to produce relative constraints between robot poses. In Pose SLAM, observations come in the form of relative-motion measurements between robot poses. With regards to extending the original Pose SLAM formulation, this thesis studies the computation of such measurements when they are obtained with stereo cameras and develops the appropriate noise propagation models for such case. Furthermore, the initial formulation of Pose SLAM assumes poses in SE(2) and in this thesis we extend this formulation to SE(3), parameterizing rotations either with Euler angles and quaternions. We also introduce a loop closure test that exploits the information from the filter using an independent measure of information content between poses. In the application domain, we present a technique to process the 3D volumetric maps obtained with this SLAM methodology, but with laser range scanning as the sensor modality, to derive traversability maps. Aside from these extensions to Pose SLAM, the core contribution of the thesis is an approach for path planning that exploits the modeled uncertainties in Pose SLAM to search for the path in the pose graph with the lowest accumulated robot pose uncertainty, i.e., the path that allows the robot to navigate to a given goal with the least probability of becoming lost. An added advantage of the proposed path planning approach is that since Pose SLAM is agnostic with respect to the sensor modalities used, it can be used in different environments and with different robots, and since the original pose graph may come from a previous mapping session, the paths stored in the map already satisfy constraints not easy modeled in the robot controller, such as the existence of restricted regions, or the right of way along paths. The proposed path planning methodology has been extensively tested both in simulation and with a real outdoor robot. Our path planning approach is adequate for scenarios where a robot is initially guided during map construction, but autonomous during execution. For other scenarios in which more autonomy is required, the robot should be able to explore the environment without any supervision. The second core contribution of this thesis is an autonomous exploration method that complements the aforementioned path planning strategy. The method selects the appropriate actions to drive the robot so as to maximize coverage and at the same time minimize localization and map uncertainties. An occupancy grid is maintained for the sole purpose of guaranteeing coverage. A significant advantage of the method is that since the grid is only computed to hypothesize entropy reduction of candidate map posteriors, it can be computed at a very coarse resolution since it is not used to maintain neither the robot localization estimate, nor the structure of the environment. Our technique evaluates two types of actions: exploratory actions and place revisiting actions. Action decisions are made based on entropy reduction estimates. By maintaining a Pose SLAM estimate at run time, the technique allows to replan trajectories online should significant change in the Pose SLAM estimate be detected. The proposed exploration strategy was tested in a common publicly available dataset comparing favorably against frontier based exploratio

Grasping and Assembling with Modular Robots

A wide variety of problems, from manufacturing to disaster response and space exploration, can benefit from robotic systems that can firmly grasp objects or assemble various structures, particularly in difficult, dangerous environments. In this thesis, we study the two problems, robotic grasping and assembly, with a modular robotic approach that can facilitate the problems with versatility and robustness. First, this thesis develops a theoretical framework for grasping objects with customized effectors that have curved contact surfaces, with applications to modular robots. We present a collection of grasps and cages that can effectively restrain the mobility of a wide range of objects including polyhedra. Each of the grasps or cages is formed by at most three effectors. A stable grasp is obtained by simple motion planning and control. Based on the theory, we create a robotic system comprised of a modular manipulator equipped with customized end-effectors and a software suite for planning and control of the manipulator. Second, this thesis presents efficient assembly planning algorithms for constructing planar target structures collectively with a collection of homogeneous mobile modular robots. The algorithms are provably correct and address arbitrary target structures that may include internal holes. The resultant assembly plan supports parallel assembly and guarantees easy accessibility in the sense that a robot does not have to pass through a narrow gap while approaching its target position. Finally, we extend the algorithms to address various symmetric patterns formed by a collection of congruent rectangles on the plane. The basic ideas in this thesis have broad applications to manufacturing (restraint), humanitarian missions (forming airfields on the high seas), and service robotics (grasping and manipulation)