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

Markus Bläser, Holger Dell, Johann A. Makowsky
2009 Theory of Computing Systems  
Our main result identifies a small, algebraic set 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  ... 
doi:10.1007/s00224-009-9213-7 fatcat:7uafgxlslffwdeswh7di6nsmjy

On the algebraic complexity of some families of coloured Tutte polynomials

Martin Lotz, Johann A. Makowsky
2004 Advances in Applied Mathematics  
Generalising 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.  ... 
doi:10.1016/s0196-8858(03)00087-3 fatcat:4ui5xgsqzffilkh4valptjqr74

The exact complexity of the Tutte polynomial [article]

Tomer Kotek, Johann A. Makowsky
2019 arXiv   pre-print
This is a survey on 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.  ... 
arXiv:1910.08915v1 fatcat:tac6owgq3fgjtchyzj3buygxlq

A Computational Framework for the Study of Partition Functions and Graph Polynomials

T. Kotek, J. A. Makowsky, E. V. Ravve
2013 Proceedings of the 12th Asian Logic Conference  
In this paper we propose a unified natural framework for 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.  ... 
doi:10.1142/9789814449274_0012 fatcat:6d46u4msrzaf7cu7uu6caikg2u

A Computational Framework for the Study of Partition Functions and Graph Polynomials

Tomer Kotek, Johann A. Makowsky, Elena V. Ravve
2012 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
In this paper we propose a unified natural framework for 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.  ... 
doi:10.1109/synasc.2012.36 dblp:conf/synasc/KotekMR12 fatcat:msge5u34prbunepemvhkuemjke

Complexity of the Bollobás-Riordan Polynomial [chapter]

Markus Bläser, Holger Dell, Johann A. Makowsky
Computer Science – Theory and Applications  
Our main result identifies a small, algebraic set 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  ... 
doi:10.1007/978-3-540-79709-8_12 dblp:conf/csr/BlaserDM08 fatcat:dzxainmgo5cjpgth3s3ww6kkja

A little statistical mechanics for the graph theorist

Laura Beaudin, Joanna Ellis-Monaghan, Greta Pangborn, Robert Shrock
2010 Discrete Mathematics  
We discuss 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] .  ... 
doi:10.1016/j.disc.2010.03.011 fatcat:roiu7evabzg3bojnbvhl56ebdm

Algorithmic uses of the Feferman–Vaught Theorem

J.A. Makowsky
2004 Annals of Pure and Applied Logic  
We then extend 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.  ... 
doi:10.1016/j.apal.2003.11.002 fatcat:mn33xk2hhvhbhdgob73g2jckeu

Combinatorics, Probability and Computing

Noga Alon, Béla Bollobás, Ingo Wegener
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 .  ... 
doi:10.4171/owr/2006/48 fatcat:cnquryyfpba6bhuesmwsx2vq34

Scale-Free Cortical Planar Networks [chapter]

Walter J. Freeman, Robert Kozma, Béla Bollobá, Oliver Riordan
2008 Bolyai Society Mathematical Studies  
theory (RGT); to choose 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 [  ... 
doi:10.1007/978-3-540-69395-6_7 fatcat:az2mktwx6jfo3c3s66gdrpdeme

Ported Tutte Functions of Extensors and Oriented Matroids [article]

Seth Chaiken
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.  ... 
arXiv:math/0605707v2 fatcat:th3pufbranahhlo73y74f3giaa

Longest Path in the Price Model [article]

Tim S. Evans, Lucille Calmon, Vaiva Vasiliauskaite
2020 arXiv   pre-print
We define a reverse greedy path 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.  ... 
arXiv:1903.03667v2 fatcat:bopqhizcpve4fiqsx4omb25dri

The history of degenerate (bipartite) extremal graph problems [article]

Zoltán Füredi, Miklós Simonovits
2013 arXiv   pre-print
This paper is a survey on Extremal Graph Theory, primarily focusing on 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.  ... 
arXiv:1306.5167v2 fatcat:t6puw44f4rayfehkdstmlorsfy

The Power of Local Information in Social Networks [chapter]

Christian Borgs, Michael Brautbar, Jennifer Chayes, Sanjeev Khanna, Brendan Lucier
2012 Lecture Notes in Computer Science  
This addresses an open question 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.  ... 
doi:10.1007/978-3-642-35311-6_30 fatcat:sjg4gr4l4vax3cghn7gooc6pjy

The complexity of symmetric boolean functions [chapter]

Ingo Wegener
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).  ... 
doi:10.1007/3-540-18170-9_185 fatcat:tmwjsublb5bohpc56dzb5zmiey
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