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Extensor-Coding [article]

Cornelius Brand, Holger Dell, Thore Husfeldt
2018 arXiv   pre-print
We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ±ε. Our algorithm runs in time ε^-2 4^k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem
more » ... nd is of independent conceptual interest. Using this approach, we also obtain a deterministic 2^k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.
arXiv:1804.09448v1 fatcat:vr5sbwdwpbbsrgxjgppwlz3him

Titelei/Inhaltsverzeichnis [chapter]

Kai Bronner, Rainer Hirt, Cornelius Ringe
2014 Audio Branding Yearbook 2013/2014  
So we have to clarify that audio branding does not equate the production of only an audio logo -it stands for a sophistic and holistic way to create and maintain audible brands.  ...  about current developments and opportunities in the audio branding industry.  ... 
doi:10.5771/9783845251875_1 fatcat:p662ravuo5eudeshwd4syejb7y

Titelei/Inhaltsverzeichnis [chapter]

Kai Bronner, Rainer Hirt, Cornelius Ringe
2015 Audio Branding Yearbook 2014/2015  
ISBN 978-3-8487-1733-0 (Print) 978-3-8452-6081-5 (ePDF) Library of Congress Cataloging-in-Publication Data Bronner, Kai / Ringe, Cornelius / Hirt, Rainer ((( ABA ))) Audio Branding Academy Yearbook 2014  ...  /2015 Kai Bronner / Cornelius Ringe / Rainer Hirt (eds.) 155 p.  ...  We want to promote the creative aspect within Audio Branding. In a convincing Audio Branding case, many fea-tures have to fit.  ... 
doi:10.5771/9783845260815-1 fatcat:6jozcqqjgzdglkpd6verko3euq

Building brand value online: exploring relationships between company and city brands

Myfanwy Trueman, Nelarine Cornelius, James Wallace, Temi Abimbola
2012 European Journal of Marketing  
Bradford Scholars -how to deposit your paper Overview Copyright check • Check if your publisher allows submission to a repository. • Use the Sherpa RoMEO database if you are not sure about your publisher's position or email openaccess@bradford.ac.uk.
doi:10.1108/03090561211230179 fatcat:rierq7vhw5fb3fdrzqbjand22e

An Algorithmic Method of Partial Derivatives [article]

Cornelius Brand, Kevin Pratt
2020 arXiv   pre-print
We study the following problem and its applications: given a homogeneous degree-d polynomial g as an arithmetic circuit, and a d × d matrix X whose entries are homogeneous linear polynomials, compute g(∂/∂ x_1, ..., ∂/∂ x_n) X. By considering special cases of this problem we obtain faster parameterized algorithms for several problems, including the matroid k-parity and k-matroid intersection problems, faster deterministic algorithms for testing if a linear space of matrices contains an
more » ... e matrix (Edmonds's problem) and detecting k-internal outbranchings, and more. We also match the runtime of the fastest known deterministic algorithm for detecting subgraphs of bounded pathwidth, while using a new approach. Our approach raises questions in algebraic complexity related to Waring rank and the exponent of matrix multiplication ω. In particular, we study a new complexity measure on the space of homogeneous polynomials, namely the bilinear complexity of a polynomial's apolar algebra. Our algorithmic improvements are reflective of the fact that for the degree-n determinant polynomial this quantity is at most O(n 2^ω n), whereas all known upper bounds on the Waring rank of this polynomial exceed n!.
arXiv:2005.05143v1 fatcat:dhvd7nftxbdw3in2gokbu2722a

Parameterized Algorithms for MILPs with Small Treedepth [article]

Cornelius Brand, Martin Koutecký, Sebastian Ordyniak
2019 arXiv   pre-print
Solving (mixed) integer linear programs, (M)ILPs for short, is a fundamental optimization task. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: n-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm
more » ... MILP in time f(a,d) poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth, and n is the number of variables. This is enabled by proving bounds on the denominators of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analyzing the inverses of invertible submatrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We trace the limiting boundary of our approach, showing that naturally related classes of linear programs have vertices of unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.
arXiv:1912.03501v1 fatcat:hibymvb4jfbvxoawiznog5flry

Parameterized counting of trees, forests and matroid bases [article]

Cornelius Brand, Marc Roth
2016 arXiv   pre-print
We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with k edges are # W[1]-hard when parameterized by k. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for k-forests implies # W[1]-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even
more » ... f the matroid is restricted to be representable over a field of characteristic 2. We complement this result by pointing out that the problem becomes fixed parameter tractable for matroids represented over a fixed finite field.
arXiv:1611.01823v1 fatcat:mzq5iqfqsrgufczdc2seww63km

Book review: Holtorf, Cornelius, Archaeology is a Brand! The Meaning of Archaeology in Contemporary Popular Culture (Oxford: Archaeopress, 2007). ix + 184pp. ISBN 9781905739066 £14.99 (paperback)

Katherine Pandora
2008 Public Understanding of Science  
Holtorf believes that they do, and in Archaeology is a Brand!  ...  In this high-spirited examination of archaeology's public image, Cornelius Holtorf sets out to illuminate current themes that mark the public understanding of archaeology as a scientific enterprisenot  ... 
doi:10.1177/09636625080170040603 fatcat:icclos5hznarjb6g74ddytczy4

Patching Colors with Tensors

Cornelius Brand, Michael Wagner
2019 European Symposium on Algorithms  
We describe a generic way of exponentially speeding up algorithms which rely on Color-Coding by using the recently introduced technique of Extensor-Coding (Brand, Dell and Husfeldt, STOC 2018).  ...  Brand 25:11 indeterminates.  ...  Brand 25:3 monomial of degree k in which no variable appears with degree more than one.  ... 
doi:10.4230/lipics.esa.2019.25 dblp:conf/esa/Brand19 fatcat:qkgawqptj5go5cz23wlxcmgwqq

Fine-grained dichotomies for the Tutte plane and Boolean #CSP [article]

Cornelius Brand, Holger Dell, Marc Roth
2016 arXiv   pre-print
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1, which was left open. We
more » ... ete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp(o(n)) unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp(o(n)) unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.
arXiv:1606.06581v1 fatcat:ut6354ay3beuva7mnmg77352bu

On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection [article]

Cornelius Brand, Michael Sagraloff
2016 arXiv   pre-print
Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The
more » ... st step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials.
arXiv:1604.08944v1 fatcat:75nlwucilfbllhna7jbw6elosa

Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Cornelius Brand, Holger Dell, Marc Roth
2018 Algorithmica  
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12] , in combination with the results of Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y = 1, which was
more » ... eft open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp o(n) unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases cannot be solved in time exp o(n) unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp o(n) unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation. Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP polynomial-time algorithm that is given access to an oracle for counting perfect matchings. This theorem suggests that counting is much harder than decision. When faced with a problem that is NP-hard or #P-hard, the area of exact algorithms strives to find the fastest exponential-time algorithm for a problem, or find reasons why faster algorithms might not exist. For example, the fastest known algorithm for counting perfect matchings in n-vertex graphs [1] runs in time 2 n/2 · poly(n). It has been hypothesized that no O 1.99 n/2 -time algorithm for the problem exists, but we do not know whether such an algorithm has implications for the strong exponential time hypothesis, which states that for all ε > 0, there is some k such that the problem of deciding satisfiability of boolean formulas in k-CNF on n variables does not have an algorithm running in time (2 − ε) n . However, we know by [8] that the term O(n) in the exponent is asymptotically tight, in the sense that a 2 o(n) -time algorithm for counting perfect matchings would violate the (randomized) exponential time hypothesis (ETH) by Impagliazzo and Paturi [13] . Using the idea of block interpolation, Curticapean [7] strengthened the hardness by showing that a 2 o(n) -time algorithm for counting perfect matchings would violate the (deterministic) counting exponential time hypothesis (#ETH). Our main results are hardness results under #ETH for 1) the problem of counting all forests in a graph, that is, its acyclic subgraphs, and 2) the problem of counting the number of independent sets in a bipartite graph. If #ETH holds, then neither of these problems has an algorithm running in time exp(o(n)) even in simple n-vertex graphs of bounded maximum degree. We use these results to lift two known "FP vs. #P-hard" dichotomy theorems to their more refined and asymptotically tight "FP vs. #ETH-hard" variants. Here FP is the class of functions computable in polynomial time. Note that #ETH is weaker than ETH, so that our results could also be stated under ETH.
doi:10.1007/s00453-018-0472-z fatcat:i2ys5fwwkngxbmy557eybmtbti

A Note on the Approximability of Deepest-Descent Circuit Steps [article]

Steffen Borgwardt, Cornelius Brand, Andreas Emil Feldmann, Martin Koutecký
2021 arXiv   pre-print
Linear programs (LPs) can be solved by polynomially many moves along the circuit direction improving the objective the most, so-called deepest-descent steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv, 2019), a consequence of the hardness of deciding the existence of an optimal circuit-neighbor (OCNP) on LPs with non-unique optima. We prove OCNP is easy under the promise of unique optima, but already O(n^1-ε)-approximating dd-steps remains hard even for totally
more » ... r n-dimensional 0/1-LPs with a unique optimum. We provide a matching n-approximation.
arXiv:2010.10809v2 fatcat:rrmdycazybb6bcwkziv3fzwv2m

Trends in the use of premium and discount cigarette brands: findings from the ITC US Surveys (2002–2011)

Monica E Cornelius, Pete Driezen, Geoffrey T Fong, Frank J Chaloupka, Andrew Hyland, Maansi Bansal-Travers, Matthew J Carpenter, K Michael Cummings
2013 Tobacco Control  
brand to another premium brand; and from a discount to premium brand.  ...  premium brands.  ... 
doi:10.1136/tobaccocontrol-2013-051045 pmid:24092600 pmcid:PMC4038075 fatcat:x3oicau6afhjrgpkow6ladl6ha

The prevalence of brand switching among adult smokers in the USA, 2006–2011: findings from the ITC US surveys

Monica E Cornelius, K Michael Cummings, Geoffrey T Fong, Andrew Hyland, Pete Driezen, Frank J Chaloupka, David Hammond, Richard J O'Connor, Maansi Bansal-Travers
2014 Tobacco Control  
Received 2 May 2014 Accepted 28 August 2014 To cite: Cornelius ABSTRACT Background Recent studies have suggested that about 1 in 5 smokers report switching brands per year.  ...  brand style.  ... 
doi:10.1136/tobaccocontrol-2014-051765 pmid:25260750 pmcid:PMC4743742 fatcat:h4xxrkcdizeg7ljlr4ob3oume4
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