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373
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 61 (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.
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 49 (15 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
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the a ..."
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Cited by 42 (5 self)
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We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10^300. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...
Frequency Assignment Problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Furt ..."
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Cited by 41 (3 self)
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The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Further at issue are online algorithms for dynamically assigning frequencies to users within an established network. Applications include aeronautical mobile, land mobile, maritime mobile, broadcast, land fixed (pointto point), and satellite systems. This paper surveys research conducted by theoreticians, engineers, and computer scientists regarding the frequency assignment problem (FAP) in all of its guises. The paper begins by defining some of the more common types of FAPs. It continues with a discussion on measures of optimality relating to the use of spectrum, models of interference, and mathematical representations of the many FAPs, both in graph theoretic terms, and as mathematical pro...
Even edge colorings of graphs
, 1985
"... It is shown that the minimum number of colors needed to paint the edges of a graph G so that in every cycle of G there is a nonzero even number of edges of at least one color is rlog,X(G)l. ..."
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Cited by 32 (1 self)
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It is shown that the minimum number of colors needed to paint the edges of a graph G so that in every cycle of G there is a nonzero even number of edges of at least one color is rlog,X(G)l.
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 32 (15 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Defective coloring revisited
 Journal of Graph Theory
, 1997
"... A graph is (k; d)colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3 ..."
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Cited by 31 (0 self)
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A graph is (k; d)colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2) and (5, 1)colorable, and that a graph of genus γ is (γ=(d + 1) + 4; d)colorable, where γ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2; k)coloring, for k 1; and the (3, 1)coloring problems are NPcomplete even for planar graphs. In general graphs (k; d)coloring is NPcomplete for k 3; d 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. c © 1997 John Wiley & Sons, Inc. 1.