4,432 research outputs found
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
A Tutorial on Clique Problems in Communications and Signal Processing
Since its first use by Euler on the problem of the seven bridges of
K\"onigsberg, graph theory has shown excellent abilities in solving and
unveiling the properties of multiple discrete optimization problems. The study
of the structure of some integer programs reveals equivalence with graph theory
problems making a large body of the literature readily available for solving
and characterizing the complexity of these problems. This tutorial presents a
framework for utilizing a particular graph theory problem, known as the clique
problem, for solving communications and signal processing problems. In
particular, the paper aims to illustrate the structural properties of integer
programs that can be formulated as clique problems through multiple examples in
communications and signal processing. To that end, the first part of the
tutorial provides various optimal and heuristic solutions for the maximum
clique, maximum weight clique, and -clique problems. The tutorial, further,
illustrates the use of the clique formulation through numerous contemporary
examples in communications and signal processing, mainly in maximum access for
non-orthogonal multiple access networks, throughput maximization using index
and instantly decodable network coding, collision-free radio frequency
identification networks, and resource allocation in cloud-radio access
networks. Finally, the tutorial sheds light on the recent advances of such
applications, and provides technical insights on ways of dealing with mixed
discrete-continuous optimization problems
AND-NOT logic framework for steady state analysis of Boolean network models
Finite dynamical systems (e.g. Boolean networks and logical models) have been
used in modeling biological systems to focus attention on the qualitative
features of the system, such as the wiring diagram. Since the analysis of such
systems is hard, it is necessary to focus on subclasses that have the
properties of being general enough for modeling and simple enough for
theoretical analysis. In this paper we propose the class of AND-NOT networks
for modeling biological systems and show that it provides several advantages.
Some of the advantages include: Any finite dynamical system can be written as
an AND-NOT network with similar dynamical properties. There is a one-to-one
correspondence between AND-NOT networks, their wiring diagrams, and their
dynamics. Results about AND-NOT networks can be stated at the wiring diagram
level without losing any information. Results about AND-NOT networks are
applicable to any Boolean network. We apply our results to a Boolean model of
Th-cell differentiation
Listing all maximal cliques in sparse graphs in near-optimal time
The degeneracy of an -vertex graph is the smallest number such that every subgraph of contains a vertex of degree at most . We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time . We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an -vertex graph with degeneracy (when is a multiple of 3 and ) is . Therefore, our algorithm matches the worst-case output size of the problem whenever
Distributed Design for Decentralized Control using Chordal Decomposition and ADMM
We propose a distributed design method for decentralized control by
exploiting the underlying sparsity properties of the problem. Our method is
based on chordal decomposition of sparse block matrices and the alternating
direction method of multipliers (ADMM). We first apply a classical
parameterization technique to restrict the optimal decentralized control into a
convex problem that inherits the sparsity pattern of the original problem. The
parameterization relies on a notion of strongly decentralized stabilization,
and sufficient conditions are discussed to guarantee this notion. Then, chordal
decomposition allows us to decompose the convex restriction into a problem with
partially coupled constraints, and the framework of ADMM enables us to solve
the decomposed problem in a distributed fashion. Consequently, the subsystems
only need to share their model data with their direct neighbours, not needing a
central computation. Numerical experiments demonstrate the effectiveness of the
proposed method.Comment: 11 pages, 8 figures. Accepted for publication in the IEEE
Transactions on Control of Network System
Mining Novel Multivariate Relationships in Time Series Data Using Correlation Networks
In many domains, there is significant interest in capturing novel
relationships between time series that represent activities recorded at
different nodes of a highly complex system. In this paper, we introduce
multipoles, a novel class of linear relationships between more than two time
series. A multipole is a set of time series that have strong linear dependence
among themselves, with the requirement that each time series makes a
significant contribution to the linear dependence. We demonstrate that most
interesting multipoles can be identified as cliques of negative correlations in
a correlation network. Such cliques are typically rare in a real-world
correlation network, which allows us to find almost all multipoles efficiently
using a clique-enumeration approach. Using our proposed framework, we
demonstrate the utility of multipoles in discovering new physical phenomena in
two scientific domains: climate science and neuroscience. In particular, we
discovered several multipole relationships that are reproducible in multiple
other independent datasets and lead to novel domain insights.Comment: This is the accepted version of article submitted to IEEE
Transactions on Knowledge and Data Engineering 201
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