18,834 research outputs found
Polynomial Time Algorithms for Tracking Path Problems
Given a graph , and terminal vertices and , the TRACKING PATHS
problem asks to compute a minimum number of vertices to be marked as trackers,
such that the sequence of trackers encountered in each s-t path is unique.
TRACKING PATHS is NP-hard in both directed and undirected graphs in general. In
this paper we give a collection of polynomial time algorithms for some
restricted versions of TRACKING PATHS. We prove that TRACKING PATHS is
polynomial time solvable for chordal graphs and tournament graphs. We prove
that TRACKING PATHS is NP-hard in graphs with bounded maximum degree
, and give a -approximate algorithm for the same. We
also analyze the version of tracking s-t paths where paths are tracked using
edges instead of vertices, and we give a polynomial time algorithm for the
same. Finally, we show how to reconstruct an s-t path, given a sequence of
trackers and a tracking set for the graph in consideration.Comment: Submitted to IWOCA 202
Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic
Numerical continuation methods track a solution path defined by a homotopy.
The systems we consider are defined by polynomials in several variables with
complex coefficients. For larger dimensions and degrees, the numerical
conditioning worsens and hardware double precision becomes often insufficient
to reach the end of the solution path. With double double and quad double
arithmetic, we can solve larger problems that we could not solve with hardware
double arithmetic, but at a higher computational cost. This cost overhead can
be compensated by acceleration on a Graphics Processing Unit (GPU). We describe
our implementation and report on computational results on benchmark polynomial
systems.Comment: Accepted for publication in the Proceedings of the 7th International
Workshop on Parallel Symbolic Computation (PASCO 2015
A Blackbox Polynomial System Solver on Parallel Shared Memory Computers
A numerical irreducible decomposition for a polynomial system provides
representations for the irreducible factors of all positive dimensional
solution sets of the system, separated from its isolated solutions. Homotopy
continuation methods are applied to compute a numerical irreducible
decomposition. Load balancing and pipelining are techniques in a parallel
implementation on a computer with multicore processors. The application of the
parallel algorithms is illustrated on solving the cyclic -roots problems, in
particular for , and~12.Comment: Accepted for publication in the proceedings of CASC 201
Numerical Algebraic Geometry for Macaulay2
Numerical Algebraic Geometry uses numerical data to describe algebraic
varieties. It is based on the methods of numerical polynomial homotopy
continuation, an alternative to the classical symbolic approaches of
computational algebraic geometry. We present a package, the driving idea behind
which is to interlink the existing symbolic methods of Macaulay2 and the
powerful engine of numerical approximate computations. The core procedures of
the package exhibit performance competitive with the other homotopy
continuation software.Comment: 7 page
A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles
Self-driving vehicles are a maturing technology with the potential to reshape
mobility by enhancing the safety, accessibility, efficiency, and convenience of
automotive transportation. Safety-critical tasks that must be executed by a
self-driving vehicle include planning of motions through a dynamic environment
shared with other vehicles and pedestrians, and their robust executions via
feedback control. The objective of this paper is to survey the current state of
the art on planning and control algorithms with particular regard to the urban
setting. A selection of proposed techniques is reviewed along with a discussion
of their effectiveness. The surveyed approaches differ in the vehicle mobility
model used, in assumptions on the structure of the environment, and in
computational requirements. The side-by-side comparison presented in this
survey helps to gain insight into the strengths and limitations of the reviewed
approaches and assists with system level design choices
Tracking Moving Objects with Few Handovers
We study the online problem of assigning a moving point to a base-station
region that contains it. For instance, the moving object could represent a
cellular phone and the base station could represent the coverage zones of cell
towers. Our goal is to minimize the number of handovers that occur when the
point moves outside its assigned region and must be assigned to a new region.
We study this problem in terms of competitive analysis and we measure the
competitive ratio of our algorithms as a function of the ply of the system of
regions, that is, the maximum number of regions that cover any single point. In
the offline version of this problem, when object motions are known in advance,
a simple greedy strategy suffices to determine an optimal assignment of objects
to base stations, with as few handovers as possible. For the online version of
this problem for moving points in one dimension, we present a deterministic
algorithm that achieves a competitive ratio of O(log ply) with respect to the
optimal algorithm, and we show that no better ratio is possible. For two or
more dimensions, we present a randomized online algorithm that achieves a
competitive ratio of O(log ply) with respect to the optimal algorithm, and a
deterministic algorithm that achieves a competitive ratio of O(ply); again, we
show that no better ratio is possible
Certified numerical homotopy tracking
Given a homotopy connecting two polynomial systems we provide a rigorous
algorithm for tracking a regular homotopy path connecting an approximate zero
of the start system to an approximate zero of the target system. Our method
uses recent results on the complexity of homotopy continuation rooted in the
alpha theory of Smale. Experimental results obtained with the implementation in
the numerical algebraic geometry package of Macaulay2 demonstrate the
practicality of the algorithm. In particular, we confirm the theoretical
results for random linear homotopies and illustrate the plausibility of a
conjecture by Shub and Smale on a good initial pair.Comment: 26 pages, 1 figur
Multiprecision path tracking
A path tracking algorithm that adaptively adjusts precision is presented. By
adjusting the level of precision in accordance with the numerical conditioning
of the path, the algorithm achieves high reliability with less computational
cost than would be incurred by raising precision across the board. We develop
simple rules for adjusting precision and show how to integrate these into an
algorithm that also adaptively adjusts the step size. The behavior of the
method is illustrated on several examples arising as homotopies for solving
systems of polynomial equations
Point trajectory planning of flexible redundant robot manipulators using genetic algorithms
The paper focuses on the problem of point-to-point trajectory planning for flexible redundant robot manipulators (FRM) in joint space. Compared with irredundant flexible manipulators, a FRM possesses additional possibilities during point-to-point trajectory planning due to its kinematics redundancy. A trajectory planning method to minimize vibration and/or executing time of a point-to-point motion is presented for FRMs based on Genetic Algorithms (GAs). Kinematics redundancy is integrated into the presented method as planning variables. Quadrinomial and quintic polynomial are used to describe the segments that connect the initial, intermediate, and final points in joint space. The trajectory planning of FRM is formulated as a problem of optimization with constraints. A planar FRM with three flexible links is used in simulation. Case studies show that the method is applicable
Efficient Algorithms for Measuring the Funnel-likeness of DAGs
Funnels are a new natural subclass of DAGs. Intuitively, a DAG is a funnel if
every source-sink path can be uniquely identified by one of its arcs. Funnels
are an analog to trees for directed graphs that is more restrictive than DAGs
but more expressive than in-/out-trees. Computational problems such as finding
vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs
while on funnels they become solvable in polynomial time. Our main focus is the
algorithmic complexity of finding out how funnel-like a given DAG is. To this
end, we study the NP-hard problem of computing the arc-deletion distance to a
funnel of a given DAG. We develop efficient exact and approximation algorithms
for the problem and test them on synthetic random graphs and real-world graphs.Comment: Submitted to ISCO 201
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