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COMPLEXITY OF GRAPH POLYNOMIALS

Steven D. Noble

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0013
Subject:
Mathematics, Probability / Statistics

This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, ... More


CONNECTIONS TO APPROXIMATION ALGORITHMS

Rolf Niedermeier

in Invitation to Fixed-Parameter Algorithms

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198566076
eISBN:
9780191713910
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566076.003.0014
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter summarizes the currently known connections between fixed-parameter and polynomial-time approximation algorithmics. Important topics in this context are lower bound and impossibility ... More


The Deep Question: P vs. NP

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the consequences of the possibility that P is equal to NP. It examines why it is extremely difficult to prove that P is not equal to NP by focusing on a set of ‘metatheorems’ ... More


Optimization and Approximation

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on the relationships between decision problems and their optimisation versions. It shows that, for most problems, the optimal solution can be realised in polynomial time if and ... More


Randomized Algorithms

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0010
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Certain situations require a random rather than a deterministic strategy. With a random strategy, the choices are unpredictable and the adversary may be kept off balance. This chapter focuses on the ... More


Counting, Sampling, and Statistical Physics

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. ... More


Computing complex zeros of polynomials and power series

Jean-Marc Couveignes

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0005
Subject:
Mathematics, Number Theory

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of ... More


Computing <i>V<sub>f</sub></i> modulo <i>p</i>

Jean-Marc Couveignes

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0013
Subject:
Mathematics, Number Theory

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing ... More


The Winner Determination Problem

Daniel Lehmann, Rudolf Müller, and Tuomas Sandholm

in Combinatorial Auctions

Published in print:
2005
Published Online:
August 2013
ISBN:
9780262033428
eISBN:
9780262302920
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262033428.003.0013
Subject:
Society and Culture, Technology and Society

This chapter defines and formulates a combinatorial optimization problem, called the winner determination problem, and examines its complexity properties. A range of alternative mathematical ... More


Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Bas Edixhoven and Jean-Marc Couveignes (eds)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.001.0001
Subject:
Mathematics, Number Theory

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's ... More


Introduction, main results, context

Bas Edixhoven

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0001
Subject:
Mathematics, Number Theory

This chapter provides an introduction to the subject, precise statements of the main results, and places these in a somewhat wider context. Topics discussed include statement of the main results, ... More


Computing the residual Galois representations

Bas Edixhoven

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0014
Subject:
Mathematics, Number Theory

This chapter proves the main result on the computation of Galois representations. It provides a detailed description of the algorithm and a rigorous proof of the complexity. It first combines the ... More


Seymour's Decomposition Theorem

James Oxley

in Matroid Theory

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198566946
eISBN:
9780191774904
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566946.003.0014
Subject:
Mathematics, Educational Mathematics

The excluded-minor theorem provides one characterization of the class of regular matroids. This chapter presents a different characterization of this class: a decomposition theorem for members of the ... More


Spectral Methods for Dimensionality Reduction

Saul Lawrence K., Weinberger Kilian Q., Sha Fei, Ham Jihun, and Lee Daniel D.

in Semi-Supervised Learning

Published in print:
2006
Published Online:
August 2013
ISBN:
9780262033589
eISBN:
9780262255899
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262033589.003.0016
Subject:
Computer Science, Machine Learning

This chapter provides an overview of unsupervised learning algorithms that can be viewed as spectral methods for linear and nonlinear dimensionality reduction. Spectral methods have recently emerged ... More


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