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The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid [article]

Andreas Björklund, Petteri Kaski
2020 arXiv   pre-print
The first design computes the Tutte polynomial of a linear matroid of dimension k on k^O(1) points in k^O(k) operations.  ...  We show that computing the Tutte polynomial of a linear matroid of dimension k on k^O(1) points over a field of k^O(1) elements requires k^Ω(k) time unless the #ETH—a counting extension of the Exponential  ...  Acknowledgments We thank the anonymous reviewers of an earlier version of this manuscript for their useful remarks and questions that led to Theorem 4.  ... 
arXiv:2003.03595v2 fatcat:txzu4d3ofve7hlu3z3sofhyaqe

Parameterized Counting of Partially Injective Homomorphisms

Marc Roth
2021 Algorithmica  
As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.  ...  the exact complexity of the subgraph counting problem (STOC 17).  ...  Furthermore the author thanks Cornelius Brand for saying "Tutte Polynomial" every once in a while, and Philip Wellnitz for providing valuable feedback on early drafts of the full version of this work.  ... 
doi:10.1007/s00453-021-00805-y fatcat:oetq4yn2nnhy7ovyr47zgch6am

Simplicial and cellular trees [chapter]

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2016 IMA Volumes in Mathematics and its Applications  
On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension.  ...  This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem  ...  Krushkal and Renardy [KR14] studied a polynomial invariant on cell complexes that generalizes the Tutte polynomial of the cellular matroid.  ... 
doi:10.1007/978-3-319-24298-9_28 fatcat:oh5q37h55jcmvady7lukhisolq

Wilsonian renormalization, differential equations and Hopf algebras [article]

Thomas Krajewski, Pierre Martinetti
2008 arXiv   pre-print
As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the q-state Potts model.  ...  Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions.  ...  Marcolli and W. van Suijlekom, who organized the conference "Combinatorics and Physics" at the MPIM in Bonn in march 2007. Work partially supported by a EU Marie Curie fellowship EIF-025947 QGNC.  ... 
arXiv:0806.4309v2 fatcat:wfhliynmgvcv3bdvssfoowetgy

Paths and walks, forests and planes : arcadian algorithms and complexity [article]

Cornelius Brand, Universität Des Saarlandes, Universität Des Saarlandes
Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways.  ...  Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis.  ...  At the danger of seeming somewhat heterodox, we collect both of these approaches under the term of fine-grained algorithms and complexity.  ... 
doi:10.22028/d291-29611 fatcat:f7r5qxb3fvegrf4hu3v5cy2zoq

Subject Index to Volumes 1–75

2001 Information Processing Letters  
, 663 domains, 1080, 1235 linear circuits, 4088 logic circuits, 751 projections, 762 resolution proofs, 1925 size of the description of a finite automaton, 663 inaccessible set, 1302 sized subset  ...  , 1790 FindMin, 1790 fine covers, 3204 grained memory model, 2411 structure, 269, 3608 finest partition, 110 topology, 1991 finger search tree, 2950 trees, 1261 finish node, 538 finishing  ...  a matrix, 3196 order n matrices, 623 tensors, 719 the adjacency matrix of a line graph, 252 rank one matrices, 1164 matrix, 472 rank queries, 1761 r decision trees, 2406 rankable set, 3245 ranked, 1610  ... 
doi:10.1016/s0020-0190(01)00175-2 fatcat:5y67tfm6yfbblakrus5nnhs73e

Dagstuhl Reports, Volume 6, Issue 6, June 2016, Complete Issue [article]

We thank all attendees of this Dagstuhl Perspectives Workshop, who have made this into a successful workshop with so many outcomes.  ...  Finally, thanks to all the staff at Dagstuhl who have made this possible. Acknowledgement. Funding by the CITEC center of excellence is gratefully acknowledged. References  ...  Tutte constructed his dichromate as a bivariate generalization of the chromatic polynomial and flow polynomial. The Tutte polynomial extends from graphs to matroids more generally.  ... 
doi:10.4230/dagrep.6.6 fatcat:bahw4gi2tjfdth3kvsi3k2uc6i

Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications [article]

Marc Roth, Universität Des Saarlandes, Universität Des Saarlandes
They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for  ...  parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational  ...  Furthermore, we will consider its data complexity which allows for a fine-grained complexity analysis.  ... 
doi:10.22028/d291-28348 fatcat:o2mlimd7fjeptaz7y2h5ffiake

Dagstuhl Reports, Volume 9, Issue 9, September 2019, Complete Issue

We describe the history of Tutte-Whitney polynomials, especially the contributions of Hassler Whitney and W. T.  ...  the Ising and Potts model partition functions of statistical physics, the weight enumerator of a linear code, and the Jones polynomial of an alternating link.  ...  Interpretations of the Tutte Polynomials of Regular Matroids Matching polynomials Complexity of evaluation of graph polynomials Question 2 We can do this in larger Hopf algebras (For examples hypergraphs  ... 
doi:10.4230/dagrep.9.9 fatcat:yxieovrp4zbxvhjyyiah6znqxm

Abstracts of Working Papers in Economics

1989 Abstracts of Working Papers in Economics  
We also describe a generalization of the Tutte polynomial for matroids to an extended Tutte polynomial for Steiner complexes.  ...  TI Matroid Steiner Problems, the Tutte Polynomial and Network Reliability. AU Colbourn, C o o k , W . P D February 1987. TI Chvatal Closures for Mixed Integer Programming Problems.  ...  We construct a "calculus for continuoustime games" that enables us to solve a certain class of timing games with a minimum of computation.  ... 
doi:10.1017/s0951007900001054 fatcat:y67mcmqgnnfhbfl34v62o5tk7a

Abstracts of Working Papers in Economics

1991 Abstracts of Working Papers in Economics  
Our estimation strategy consists of specifying the wage as a function of firm-specific value added per worker instrumented with the price of imports and the price of exports in the industry.  ...  We exploit the fact that in a small open economy such as Canada the price of imports and exports should represent pure demand shocks.  ...  We also show the existence, for any n, of a family of matrices M for which the number of shortest vectors has as a lower bound a polynomial in the bit size of M of the same degree as the polynomial bound  ... 
doi:10.1017/s0951007900002497 fatcat:rdh4eij2bbfrjgnmqxjdawyryi

Dagstuhl Reports, Volume 7, Issue 1, January 2017, Complete Issue [article]

We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems.  ...  Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time.  ...  Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph.  ... 
doi:10.4230/dagrep.7.1 fatcat:eb5yydhkhnb4xpbvalztqllpt4

Dagstuhl Reports, Volume 6, Issue 11, November 2016, Complete Issue [article]

The talk summarized the development of kernel methods for machine learning, focusing on the main ideas and some future possibilities: introduction of p.d. kernels within the theory of integral equations  ...  , their use in potential functions methods, in SVM, the general "kernel trick", the observation that kernels can be defined on arbitrary sets of objects, the link to GPs, and finally the idea to represent  ...  In this talk, I have therefore introduced a simple fine-grained cascading model, inspired from the St.  ... 
doi:10.4230/dagrep.6.11 fatcat:tfkdfittpjdydfv7ejvk4bvnh4

Abstracts of Working Papers in Economics

1990 Abstracts of Working Papers in Economics  
The implications of a fixed rate EMS with German leadership are compared with those of a cooperative fixed exchange rate regime.  ...  AB In recent years a number of structural changes in the international oil market have taken place, including the increasing importance of the spot market vis-a-vis OPEC in determining oil prices.  ...  The rank function of a pseudomatroid or delta matroid or metroid is a directed bisubmodular function.  ... 
doi:10.1017/s0951007900001613 fatcat:5qfw3qyg75gjtb2qwmceqprqsi

2016 Jahresbericht Annual Report

Jähnichen Stefan, Raimund Geschäftsführung, Seidel, Meißner Heike, Gesellschafter
The organisers would like to express their gratitude to the participants and the Schloss Dagstuhl team for a productive and exciting seminar.  ...  We organizers are all thankful to the participants, who all brought a unique insight to the seminar, which in my humble opinion, succeeded in its aims.  ...  The polynomial overhead is fine for algorithms running in polynomial time.  ...