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Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions

2009
*
Theory of Computing Systems
*

Our main result identifies a small, algebraic set

doi:10.1007/s00224-009-9213-7
fatcat:7uafgxlslffwdeswh7di6nsmjy
*of**exceptional**points**and*says that*the*evaluation problem*of**the*coloured Tutte is equivalent for all non-*exceptional**points*, under*polynomial*-time*uniform*...*The*coloured Tutte*polynomial*by*Bollobás**and**Riordan*is, as a generalization*of**the*Tutte*polynomial*,*the*most general graph*polynomial*for coloured graphs that satisfies certain contraction-deletion ...*Complexity**of**the**Bollobás*-*Riordan**Polynomial**The*dichotomy theorem from*the*classical paper [15] says that evaluating*the*Tutte*polynomial*over R is #P-hard,*except*for*the*special*points*(1, 1), (0 ...##
###
On the algebraic complexity of some families of coloured Tutte polynomials

2004
*
Advances in Applied Mathematics
*

Generalising

doi:10.1016/s0196-8858(03)00087-3
fatcat:4ui5xgsqzffilkh4valptjqr74
*the*well-known relationship between*the*Tutte*polynomial**and**the*partition function from*the*Ising model, we establish a*reduction*from*the*permanent to*the*coloured Tutte*polynomial*, thus ... We investigate*the*coloured Tutte*polynomial*in Valiant's algebraic framework*of*NP-completeness. ... We also thank*the*anonymous referee for some useful comments*and*suggestions. ...##
###
The exact complexity of the Tutte polynomial
[article]

2019
*
arXiv
*
pre-print

This is a survey on

arXiv:1910.08915v1
fatcat:tac6owgq3fgjtchyzj3buygxlq
*the*exact*complexity**of*computing*the*Tutte*polynomial*. It is*the*longer 2017 version*of*Chapter 25*of**the*CRC Handbook on*the*Tutte*polynomial**and*related topics, edited by J. ... Ellis-Monaghan*and*I. Moffatt, which is due to appear in*the*first quarter*of*2020. In*the*version to be published in*the*Handbook*the*Sections 5*and*6 are shortened*and*made into a single section. ... This was rediscovered by*Bollobás**and**Riordan*[18] as colored versions*of**the*Tutte*polynomial*. Theorem 8 (I. Averbouch, B. Godlin*and*J. ...##
###
A Computational Framework for the Study of Partition Functions and Graph Polynomials

2013
*
Proceedings of the 12th Asian Logic Conference
*

In this paper we propose a unified natural framework for

doi:10.1142/9789814449274_0012
fatcat:6d46u4msrzaf7cu7uu6caikg2u
*the*study*of*computability*and**complexity**of*partition functions*and*graph*polynomials**and*show how classical results can be cast in this framework ... Partition functions*and*graph*polynomials*have found many applications in combinatorics, physics, biology*and*even*the*mathematics*of*finance. Studying their*complexity*poses some problems. ... Acknowledgment*The*authors would like to thank I. Averbouch, P. Bürgisser, K. Meer, P. Koiran,*and*an anonymous referee for useful comments on preliminary versions*of*this paper. ...##
###
A Computational Framework for the Study of Partition Functions and Graph Polynomials

2012
*
2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
*

In this paper we propose a unified natural framework for

doi:10.1109/synasc.2012.36
dblp:conf/synasc/KotekMR12
fatcat:msge5u34prbunepemvhkuemjke
*the*study*of*computability*and**complexity**of*partition functions*and*graph*polynomials**and*show how classical results can be cast in this framework ... Partition functions*and*graph*polynomials*have found many applications in combinatorics, physics, biology*and*even*the*mathematics*of*finance. Studying their*complexity*poses some problems. ... Acknowledgment*The*authors would like to thank I. Averbouch, P. Bürgisser, K. Meer, P. Koiran,*and*an anonymous referee for useful comments on preliminary versions*of*this paper. ...##
###
Complexity of the Bollobás-Riordan Polynomial
[chapter]

*
Computer Science – Theory and Applications
*

Our main result identifies a small, algebraic set

doi:10.1007/978-3-540-79709-8_12
dblp:conf/csr/BlaserDM08
fatcat:dzxainmgo5cjpgth3s3ww6kkja
*of**exceptional**points**and*says that*the*evaluation problem*of**the*coloured Tutte is equivalent for all non-*exceptional**points*, under*polynomial*-time*uniform*...*The*coloured Tutte*polynomial*by*Bollobás**and**Riordan*is, as a generalization*of**the*Tutte*polynomial*,*the*most general graph*polynomial*for coloured graphs that satisfies certain contraction-deletion ...*Complexity**of**the**Bollobás*-*Riordan**Polynomial**The*dichotomy theorem from*the*classical paper [10] says that evaluating*the*Tutte*polynomial*over R is #P-hard,*except*for*the*special*points*(1, 1), (0 ...##
###
A little statistical mechanics for the graph theorist

2010
*
Discrete Mathematics
*

We discuss

doi:10.1016/j.disc.2010.03.011
fatcat:roiu7evabzg3bojnbvhl56ebdm
*the*equivalence*of**the*chromatic*polynomial**and**the*zero-temperature antiferromagnetic partition function,*and*how this has led to*the*study*of**the**complex*zeros*of*these functions. ... We present*the*surprising equivalence*of**the*Potts model partition function*and*one*of**the*most renowned graph invariants,*the*Tutte*polynomial*, a relationship that has resulted in a remarkable synergy ... See Traldi [141] , Zaslavsky [167] ,*Bollobás**and**Riordan*[23] ,*and*Ellis-Monaghan*and*Traldi [55] . ...##
###
Algorithmic uses of the Feferman–Vaught Theorem

2004
*
Annals of Pure and Applied Logic
*

We then extend

doi:10.1016/j.apal.2003.11.002
fatcat:mn33xk2hhvhbhdgob73g2jckeu
*the*technique to graph*polynomials*where*the*range*of**the*summation*of**the*monomials is deÿnable in MSOL. ... to*the*computation*of*truth values*of*other ÿrst order sentences in*the*factors*and*evaluation*of*a monadic second order sentence in*the*index structure. ... Recently,*Bollobas**and**Riordan*[13] have introduced a generalization*of**the*Tutte*polynomial*to edge coloured graphs,*the*coloured Tutte*polynomial*. ...##
###
Combinatorics, Probability and Computing

2006
*
Oberwolfach Reports
*

*and*random triangulations, on

*the*design

*and*analysis

*of*randomized algorithms,

*and*on

*the*relationship between

*complexity*

*and*randomness. ...

*The*probabilistic

*point*

*of*view has turned out to be very profitable in Discrete Mathematics, Analysis

*and*Theoretical Computer Science. ...

*Bollobás*, Janson

*and*

*Riordan*showed that for their model,

*the*critical

*point*is determined by a certain multi-type branching process associated to W . ...

##
###
Scale-Free Cortical Planar Networks
[chapter]

2008
*
Bolyai Society Mathematical Studies
*

theory (RGT); to choose

doi:10.1007/978-3-540-69395-6_7
fatcat:az2mktwx6jfo3c3s66gdrpdeme
*the*levels*of*description*and*scales in*the*hierarchy*of*neurodynamics; to define an appropriate module for each level;*and*to address questions*of*boundary conditions, linearity ... RGT serves to model criticality*and**the*phase transition*and**the*basic operations*of*perception in three-layered allocortex. ... Many other models soon followed, for example*the*much more general (but harder to analyze) model*of*Cooper*and*Frieze [31] ,*and**the*directed scale-free model*of**Bollobás*, Borgs, Chayes*and**Riordan*[ ...##
###
Ported Tutte Functions of Extensors and Oriented Matroids
[article]

2006
*
arXiv
*
pre-print

*The*corank-nullity

*polynomial*, basis expansions with activities,

*and*a geometric lattice expansion generalize to ported Tutte functions

*of*oriented matroids. ...

*The*Tutte equations are ported (or set-

*pointed*) when

*the*equations F(N) = g_e F(N/e) + r_e F(N\e) are omitted for elements e in a distinguished set called ports. ... Stanley, Lorenzo Traldi

*and*David G. Wagner for helpful discussions, correspondence

*and*encouragement;

*and*Thomas Zaslavsky for his assistance leading to a much improved presentation

*of*

*the*subject. ...

##
###
Longest Path in the Price Model
[article]

2020
*
arXiv
*
pre-print

We define a reverse greedy path

arXiv:1903.03667v2
fatcat:bopqhizcpve4fiqsx4omb25dri
*and*show both analytically*and*numerically that this scales with*the*logarithm*of**the*size*of**the*network with a coefficient given by*the*number*of*edges added using random ... This is a lower bound on*the*length*of**the*longest path to any given vertex*and*we show numerically that*the*longest path also scales with*the*logarithm*of**the*size*of**the*network but with a larger coefficient ... Author contributions statement T.S.E. derived*the*analytical solution. L.C.*and*V.V. conducted*the*data analysis. T.S.E.*and*V.V. wrote*the*manuscript. ...##
###
The history of degenerate (bipartite) extremal graph problems
[article]

2013
*
arXiv
*
pre-print

This paper is a survey on Extremal Graph Theory, primarily focusing on

arXiv:1306.5167v2
fatcat:t6puw44f4rayfehkdstmlorsfy
*the*case when one*of**the*excluded graphs is bipartite. ... On one hand we give an introduction to this field*and*also describe many important results, methods, problems,*and*constructions. ... Acknowledgements*The*authors are greatly indebted for fruitful discussions*and*helps to a great number*of*colleagues, among others to R. Faudree, E. Győri,*and*Z. Nagy. ...##
###
The Power of Local Information in Social Networks
[chapter]

2012
*
Lecture Notes in Computer Science
*

This addresses an open question

doi:10.1007/978-3-642-35311-6_30
fatcat:sjg4gr4l4vax3cghn7gooc6pjy
*of**Bollobás**and**Riordan*. ... We study*the*power*of*local information algorithms for optimization problems on social*and*technological networks. ... Acknowledgments*The*author Sanjeev Khanna was supported in part by NSF awards CCF-1116961*and*IIS-0904314,*and*by ONR MURI grant N00014-08-1-0747. ...##
###
The complexity of symmetric boolean functions
[chapter]

1987
*
Lecture Notes in Computer Science
*

*The*earlier papers

*of*Shannon (38)

*and*

*Riordan*

*and*Shannon (42) should also be cited. I tried to mention

*the*most relevant papers on

*the*

*complexity*

*of*Boolean functions. ... Relationships between various parameters

*of*

*complexity*

*and*various models are studied,

*and*also

*the*relationships to

*the*theory

*of*

*complexity*

*and*

*uniform*computation models. ... This lower bound method has been introduced by Wegener (84 c) for

*the*proof

*of*lower bounds on

*the*BP1-

*complexity*

*of*clique functions cl n k (see Def. 11.1, Ch. 6). ...

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