121 research outputs found
Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs
A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums
We introduce the method of path-sums which is a tool for exactly evaluating a
function of a discrete matrix with possibly non-commuting entries, based on the
closed-form resummation of infinite families of terms in the corresponding
Taylor series. If the matrix is finite, our approach yields the exact result in
a finite number of steps. We achieve this by combining a mapping between matrix
powers and walks on a weighted directed graph with a universal graph-theoretic
result on the structure of such walks. We present path-sum expressions for a
matrix raised to a complex power, the matrix exponential, matrix inverse, and
matrix logarithm. We show that the quasideterminants of a matrix can be
naturally formulated in terms of a path-sum, and present examples of the
application of the path-sum method. We show that obtaining the inversion height
of a matrix inverse and of quasideterminants is an NP-complete problem.Comment: 23 pages, light version submitted to SIAM Journal on Matrix Analysis
and Applications (SIMAX). A separate paper with the graph theoretic results
is available at: arXiv:1202.5523v1. Results for matrices over division rings
will be published separately as wel
New algebraic relationships between tight binding models
In this thesis, we present a new perspective on tight binding models. Utilising the rich algebraic toolkit provided by a combination of graph and matrix theory allows us to explore tight binding systems related through polynomial relationships. By utilising ring operations of weighted digraphs through intermediate König digraph representations, we establish a polynomial algebra over finite and infinite periodic graphs, analogous to polynomial operations on adjacency matrices. Exploring the microscopic and macroscopic behaviour of polynomials in a graph-theoretic setting, we reveal elegant relationships between the symmetrical, topological, and spectral properties of a parent graph G and its family of child graphs p(G). Drawing a correspondence between graphs and tight binding models, we investigate deep-rooted connections between different quantum systems, providing a fresh angle from which to view established tight binding models. Finally, we visit topological chains, demonstrate how their properties relate to more trivial underlying chains through effective “square root” operations, and provide new insights into their spectral characteristics
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
- …