15,966 research outputs found
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
Scaling hypothesis for the Euclidean bipartite matching problem
We propose a simple yet very predictive form, based on a Poisson's equation,
for the functional dependence of the cost from the density of points in the
Euclidean bipartite matching problem. This leads, for quadratic costs, to the
analytic prediction of the large limit of the average cost in dimension
and of the subleading correction in higher dimension. A non-trivial
scaling exponent, , which differs from the
monopartite's one, is found for the subleading correction. We argue that the
same scaling holds true for a generic cost exponent in dimension .Comment: 11 page
A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks
Research Report UPC-DEIO DR 2018-01. November 2018The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination
of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows problems in bipartite networks, for convex objective functions with diagonal Hessians (i.e., either linear, quadratic or separable nonlinear objectives). After analyzing the theoretical properties of the interior-point method for this kind of problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. For linear and quadratic instances our approach is compared with the barriers algorithms of CPLEX (both standard path-following and homogeneous-self-dual); for linear instances it is also compared with the different algorithms of the state-of-the-art network flow solver LEMON (namely: network simplex, capacity scaling, cost scaling and cycle canceling).
The specialized interior-point approach significantly outperformed the other approaches in most of the linear and quadratic transportation instances tested. In particular, it always provided a solution within the time limit and it never exhausted the 192 Gigabytes of memory of the server used for the runs. For assignment problems the network algorithms in LEMON were the most efficient option.Peer ReviewedPreprin
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a
shorter version of this article appeared in the April 20, 2001 issue of
Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
Several recent methods used to analyze asymptotic stability of
delay-differential equations (DDEs) involve determining the eigenvalues of a
matrix, a matrix pencil or a matrix polynomial constructed by Kronecker
products. Despite some similarities between the different types of these
so-called matrix pencil methods, the general ideas used as well as the proofs
differ considerably. Moreover, the available theory hardly reveals the
relations between the different methods.
In this work, a different derivation of various matrix pencil methods is
presented using a unifying framework of a new type of eigenvalue problem: the
polynomial two-parameter eigenvalue problem, of which the quadratic
two-parameter eigenvalue problem is a special case. This framework makes it
possible to establish relations between various seemingly different methods and
provides further insight in the theory of matrix pencil methods.
We also recognize a few new matrix pencil variants to determine DDE
stability.
Finally, the recognition of the new types of eigenvalue problem opens a door
to efficient computation of DDE stability
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