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Extensor-Coding
[article]

2018
*
arXiv
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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

arXiv:1804.09448v1
fatcat:vr5sbwdwpbbsrgxjgppwlz3him
## 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.##
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Titelei/Inhaltsverzeichnis
[chapter]

2014
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Audio Branding Yearbook 2013/2014
*

So we have to clarify that audio

doi:10.5771/9783845251875_1
fatcat:p662ravuo5eudeshwd4syejb7y
*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. ...##
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Titelei/Inhaltsverzeichnis
[chapter]

2015
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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,

doi:10.5771/9783845260815-1
fatcat:6jozcqqjgzdglkpd6verko3euq
*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. ...##
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Building brand value online: exploring relationships between company and city brands

2012
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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
##
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An Algorithmic Method of Partial Derivatives
[article]

2020
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arXiv
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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

arXiv:2005.05143v1
fatcat:dhvd7nftxbdw3in2gokbu2722a
## 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!.##
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Parameterized Algorithms for MILPs with Small Treedepth
[article]

2019
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arXiv
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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

arXiv:1912.03501v1
fatcat:hibymvb4jfbvxoawiznog5flry
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... 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.##
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Parameterized counting of trees, forests and matroid bases
[article]

2016
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arXiv
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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

arXiv:1611.01823v1
fatcat:mzq5iqfqsrgufczdc2seww63km
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... 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.##
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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)

2008
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Public Understanding of Science
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Holtorf believes that they do, and in Archaeology is a

doi:10.1177/09636625080170040603
fatcat:icclos5hznarjb6g74ddytczy4
*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 ...##
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Patching Colors with Tensors

2019
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European Symposium on Algorithms
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We describe a generic way of exponentially speeding up algorithms which rely on Color-Coding by using the recently introduced technique of Extensor-Coding (

doi:10.4230/lipics.esa.2019.25
dblp:conf/esa/Brand19
fatcat:qkgawqptj5go5cz23wlxcmgwqq
*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. ...##
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Fine-grained dichotomies for the Tutte plane and Boolean #CSP
[article]

2016
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arXiv
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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

arXiv:1606.06581v1
fatcat:ut6354ay3beuva7mnmg77352bu
## 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.##
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On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
[article]

2016
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arXiv
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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

arXiv:1604.08944v1
fatcat:75nlwucilfbllhna7jbw6elosa
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... 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.##
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Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

2018
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Algorithmica
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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

doi:10.1007/s00453-018-0472-z
fatcat:i2ys5fwwkngxbmy557eybmtbti
## 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.##
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A Note on the Approximability of Deepest-Descent Circuit Steps
[article]

2021
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arXiv
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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

arXiv:2010.10809v2
fatcat:rrmdycazybb6bcwkziv3fzwv2m
## more »

... r n-dimensional 0/1-LPs with a unique optimum. We provide a matching n-approximation.##
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Trends in the use of premium and discount cigarette brands: findings from the ITC US Surveys (2002–2011)

2013
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Tobacco Control
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*brand*to another premium

*brand*; and from a discount to premium

*brand*. ... premium

*brands*. ...

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

2014
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Tobacco Control
*

Received 2 May 2014 Accepted 28 August 2014 To cite:

doi:10.1136/tobaccocontrol-2014-051765
pmid:25260750
pmcid:PMC4743742
fatcat:h4xxrkcdizeg7ljlr4ob3oume4
*Cornelius*ABSTRACT Background Recent studies have suggested that about 1 in 5 smokers report switching*brands*per year. ...*brand*style. ...
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