31,869 research outputs found
A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs
In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version
An efficient method for multiobjective optimal control and optimal control subject to integral constraints
We introduce a new and efficient numerical method for multicriterion optimal
control and single criterion optimal control under integral constraints. The
approach is based on extending the state space to include information on a
"budget" remaining to satisfy each constraint; the augmented
Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our
approach hinges on the causality in that PDE, i.e., the monotonicity of
characteristic curves in one of the newly added dimensions. A semi-Lagrangian
"marching" method is used to approximate the discontinuous viscosity solution
efficiently. We compare this to a recently introduced "weighted sum" based
algorithm for the same problem. We illustrate our method using examples from
flight path planning and robotic navigation in the presence of friendly and
adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures.
Since the previous version: typos fixed, formatting improved, one mistake in
bibliography correcte
Finding tight Hamilton cycles in random hypergraphs faster
In an -uniform hypergraph on vertices a tight Hamilton cycle consists
of edges such that there exists a cyclic ordering of the vertices where the
edges correspond to consecutive segments of vertices. We provide a first
deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton
cycles in random -uniform hypergraphs with edge probability at least . Our result partially answers a question of Dudek and Frieze [Random
Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton
cycles exists already for for and for
using a second moment argument. Moreover our algorithm is superior to
previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures
& Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c
[arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a
randomised polynomial time algorithm working for edge probabilities , while the algorithm of Nenadov and \v{S}kori\'c is a
randomised quasipolynomial time algorithm working for edge probabilities .Comment: 17 page
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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