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39
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 39 (7 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.
Acyclic and Oriented Chromatic Numbers of Graphs
 J. Graph Theory
, 1997
"... . The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic n ..."
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Cited by 39 (13 self)
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. The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for o (G) in terms of a (G). An upper bound for o (G) in terms of a (G) was given by Raspaud and Sopena. We also give an upper bound for o (G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. Keywords. Oriented chromatic number, Acyclic chromatic number. 1
Layout of Graphs with Bounded TreeWidth
 2002, submitted. Stacks, Queues and Tracks: Layouts of Graph Subdivisions 41
, 2004
"... A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a gr ..."
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Cited by 25 (19 self)
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a graph represents the vertices by points in Z and the edges by noncrossing linesegments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of threedimensional drawing of a graph G is closely related to the queuenumber of G. In particular, if G is an nvertex member of a proper minorclosed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queuenumber.
Star Coloring of Graphs
, 2001
"... A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star ..."
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Cited by 25 (1 self)
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A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star
A New Algorithmic Approach to the General Lovász Local Lemma with Applications to Scheduling and Satisfiability Problems (Extended Abstract)
 STOC
, 2000
"... The LovAsz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. It has led to solutions for numerous problems in many different areas, reaching from problems in pure combinatorics to problems in routing, scheduling and approximation theory. However ..."
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Cited by 16 (2 self)
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The LovAsz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. It has led to solutions for numerous problems in many different areas, reaching from problems in pure combinatorics to problems in routing, scheduling and approximation theory. However, since the original lemma is nonconstructive, many of these solutions were first purely existential. A breakthrough result by Beck and its generalizations have led to polynomial time algorithms for many ~f these problems. However, these methods can only be applied to a simple, symmetric form of the LLL. In this paper we provide a novel approach to design polynomialtime algorithms for problems that require the LLL in its general form. We apply our techniques to find good approximate solutions to a large class of NPhard problems called minimax integer programs (MIPs). Our method finds approximate solutions that are especially for problems of nonuniform character significantly better than all methods presented before. To demonstrate the applicability of our approach, we apply it to transform important results in the area of job shop scheduling that have so far been only existential (due to the fact that the general LLL was used) into algorithms that find the predicted solutions (with only a small loss) in polynomial time. Furthermore,
Acyclic edge colorings of graphs
 Journal of Graph Theory
, 2001
"... Abstract: A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum d ..."
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Cited by 12 (1 self)
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Abstract: A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum degree in G. It is known that a 0 (G) 16 D(G) for any graph G. We prove that ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ
Coloring Nonuniform Hypergraphs: A New Algorithmic Approach to the General Lovász Local Lemma
"... The Lovász Local Lemma (LLL) is a sieve method to prove the existence of certain structures with certain prescribed properties. In most of its applications the LLL does not supply a polynomialtime algorithm for finding these structures. Beck was the first who gave a method of converting some of the ..."
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Cited by 12 (3 self)
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The Lovász Local Lemma (LLL) is a sieve method to prove the existence of certain structures with certain prescribed properties. In most of its applications the LLL does not supply a polynomialtime algorithm for finding these structures. Beck was the first who gave a method of converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. He applied his technique to the symmetric form of the LLL and, in particular, to the problem of 2coloring uniform hypergraphs. In this paper we investigate the general form of the LLL. Our main result is a randomized algorithm for 2coloring nonuniform hypergraphs that runs in expected linear time. Even for uniform hypergraphs, no algorithm with such a runtime bound was previously known, and no polynomialtime algorithm was known at all for the class of nonuniform hypergraphs we will consider in this paper. Our algorithm and its analysis provide a novel approach to the general LLL that may be of independent interest. We also show how to extend our result to the ccoloring problem.
Coloring with no 2colored P4's
, 2004
"... A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that ..."
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Cited by 12 (0 self)
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A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that
The Acyclic Edge Chromatic Number of a Random dRegular Graph is d + 1
, 2001
"... We prove the theorem from the title: the acyclic edge chromatic number of a random dregular graph is asymptotically almost surely ..."
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Cited by 11 (1 self)
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We prove the theorem from the title: the acyclic edge chromatic number of a random dregular graph is asymptotically almost surely
Algorithmic aspects of acyclic edge colorings
 Algorithmica
"... A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a ′ (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a ′ (G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in ..."
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Cited by 10 (0 self)
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A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a ′ (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a ′ (G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a ′ (G) ≤ ∆ + 2 for almost all ∆regular graphs, including all ∆regular graphs whose girth is at least c ∆ log ∆. We prove that determining the acyclic edge chromatic number of an arbitrary graph is an NPcomplete problem. For graphs G with sufficiently large girth in terms of ∆(G), we present deterministic polynomial time algorithms that color the edges of G acyclically using at most ∆(G) + 2 colors. 1