96,834 research outputs found
Levelable Sets and the Algebraic Structure of Parameterizations
Asking which sets are fixed-parameter tractable for a given parameterization
constitutes much of the current research in parameterized complexity theory.
This approach faces some of the core difficulties in complexity theory. By
focussing instead on the parameterizations that make a given set
fixed-parameter tractable, we circumvent these difficulties. We isolate
parameterizations as independent measures of complexity and study their
underlying algebraic structure. Thus we are able to compare parameterizations,
which establishes a hierarchy of complexity that is much stronger than that
present in typical parameterized algorithms races. Among other results, we find
that no practically fixed-parameter tractable sets have optimal
parameterizations
Learning Equilibria with Partial Information in Decentralized Wireless Networks
In this article, a survey of several important equilibrium concepts for
decentralized networks is presented. The term decentralized is used here to
refer to scenarios where decisions (e.g., choosing a power allocation policy)
are taken autonomously by devices interacting with each other (e.g., through
mutual interference). The iterative long-term interaction is characterized by
stable points of the wireless network called equilibria. The interest in these
equilibria stems from the relevance of network stability and the fact that they
can be achieved by letting radio devices to repeatedly interact over time. To
achieve these equilibria, several learning techniques, namely, the best
response dynamics, fictitious play, smoothed fictitious play, reinforcement
learning algorithms, and regret matching, are discussed in terms of information
requirements and convergence properties. Most of the notions introduced here,
for both equilibria and learning schemes, are illustrated by a simple case
study, namely, an interference channel with two transmitter-receiver pairs.Comment: 16 pages, 5 figures, 1 table. To appear in IEEE Communication
Magazine, special Issue on Game Theor
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
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