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368
The FourColour Theorem
, 1997
"... The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken’s in several respects. ..."
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Cited by 160 (14 self)
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The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken’s in several respects.
Finite permutation groups and finite simple groups
 Bull. London Math. Soc
, 1981
"... In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been ..."
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Cited by 127 (4 self)
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In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of
A Compiled Implementation of Strong Reduction
"... Motivated by applications to proof assistants based on dependent types, we develop and prove correct a strong reducer and b equivalence checker for the lcalculus with products, sums, and guarded fixpoints. Our approach is based on compilation to the bytecode of an abstract machine performing weak ..."
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Cited by 92 (5 self)
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Motivated by applications to proof assistants based on dependent types, we develop and prove correct a strong reducer and b equivalence checker for the lcalculus with products, sums, and guarded fixpoints. Our approach is based on compilation to the bytecode of an abstract machine performing weak reductions on nonclosed terms, derived with minimal modifications from the ZAM machine used in the Objective Caml bytecode interpreter, and complemented by a recursive "read back" procedure. An implementation in the Coq proof assistant demonstrates important speedups compared with the original interpreterbased implementation of strong reduction in Coq.
Approximating the minimumdegree Steiner tree to within one of optimal
 JOURNAL OF ALGORITHMS
, 1994
"... ... some optimal tree for the respective problems. Unless P = N P, this is the best bound achievable in polynomial time. ..."
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Cited by 88 (6 self)
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... some optimal tree for the respective problems. Unless P = N P, this is the best bound achievable in polynomial time.
Cell Population Tracking and Lineage Construction with Spatiotemporal Context
, 2009
"... Automated visualtracking of cell populations in vitro using timelapse phase contrast microscopy enables quantitative, systematic and highthroughput measurements of cell behaviors. These measurements include the spatiotemporal quantification of cell migration, mitosis, apoptosis, and the reconstru ..."
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Cited by 76 (10 self)
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Automated visualtracking of cell populations in vitro using timelapse phase contrast microscopy enables quantitative, systematic and highthroughput measurements of cell behaviors. These measurements include the spatiotemporal quantification of cell migration, mitosis, apoptosis, and the reconstruction of cell lineages. The combination of low signaltonoise ratio of phase contrast microscopy images, high and varying densities of the cell cultures, topological complexities of cell shapes, and wide range of cell behaviors poses many challenges to existing tracking techniques. This paper presents a fullyautomated multitarget tracking system that can efficiently cope with these challenges while simultaneously tracking and analyzing thousands of cells observed using timelapse phase contrast microscopy. The system combines bottomup and topdown image analysis by integrating multiple collaborative modules, which exploit a fast geometric active contour tracker in conjunction with adaptive interacting multiple models (IMM) motion filtering and spatiotemporal trajectory optimization. The system, which was tested using a variety of cell populations, achieved tracking accuracy in the range of 86.9%92.5%.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
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Cited by 61 (14 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specific ..."
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Cited by 59 (11 self)
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Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 58 (15 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
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Cited by 58 (2 self)
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Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 54 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.