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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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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.
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 (straightline grid) drawing of a gra ..."
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Cited by 31 (23 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 (straightline 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.
Nonrepetitive colorings of graphs of bounded treewidth, manuscript
, 2003
"... A sequence of the form s1s2...sms1s2...sm is called a repetition. A vertexcoloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [5] by proving that every outerplanar graph has a nonrepetitive 12coloring. We also show that graphs ..."
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Cited by 24 (0 self)
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A sequence of the form s1s2...sms1s2...sm is called a repetition. A vertexcoloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [5] by proving that every outerplanar graph has a nonrepetitive 12coloring. We also show that graphs of treewidth t have nonrepetitive 4 tcolorings. 1
New acyclic and star coloring algorithms with application to computing Hessians
 SIAM JOURNAL ON SCIENTIFIC COMPUTING VOL
, 2007
"... Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions fo ..."
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Cited by 20 (14 self)
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Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions for these two NPhard problems. The underlying common technique is the exploitation of the structure of twocolored induced subgraphs. For a graph G on n vertices and m edges, the time complexity of our star coloring algorithm is O(nd2), where dk, a generalization of vertex degree, denotes the average number of distinct paths of length at most k edges starting at a vertex in G. The time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of Ackermann’s function. The space complexity of both algorithms is O(m). To the best of our knowledge, our work is the first practical algorithm for the acyclic coloring problem. For the star coloring problem, our algorithm uses fewer colors and is considerably faster than a previously known O(nd3)time algorithm. Computational results from experiments on various largesize test graphs demonstrate that the algorithms are fast and produce highly effective solutions. The use of these algorithms in Hessian computation is expected to reduce overall runtime drastically.
Acyclic Coloring of Graphs of Maximum Degree ∆
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We sho ..."
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Cited by 5 (0 self)
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An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We show that any graph of maximum degree ∆ has acyclic chromatic number at most ∆(∆−1) 2 for any ∆ ≥ 5, and we give an O(n ∆ 2) algorithm to acyclically color any graph of maximum degree ∆ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding a(G) ≤ ∆( ∆ − 1) + 2 [ACK + 04]. By a deeper study of the case ∆ = 5, we also show that any graph of maximum degree 5 can be acyclically colored with at most 9 colors, and give a linear time algorithm to achieve this bound.
Star coloring of sparse graphs
"... A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chr ..."
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A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χs(G) and the maximum average degree Mad(G) of a graph G. We prove that: 1. If G is a graph with Mad(G) < 26, then χs(G) ≤ 4.
Graphs with maximum degree 5 are acyclically 7colorable
, 2011
"... An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + b (r+1)24 ..."
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Cited by 3 (0 self)
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An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + b (r+1)24 c)coloring.
To cite this version:
, 2011
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 1 (0 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Restricted coloring problems on graphs with few P ′4s
NONREPETITIVE COLOURING VIA ENTROPY COMPRESSION
, 2013
"... A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively `choosable if given lists of at least ` colours at each vertex, there is a nonrepetitive colouring such that each vertex is colour ..."
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A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively `choosable if given lists of at least ` colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that, for some constant c, every graph with maximum degree ∆ is c∆2choosable. We prove this result with c=1 (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5choosable. The proofs of both these results are based on the MoserTardos entropycompression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that graphs with pathwidth θ are nonrepetitively O(θ2)colourable. 1.
On Vertex Rankings of Graphs and its Relatives
"... A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should ..."
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A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length l for some fixed l. For instance, already the case l = 2 exhibits quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs excluding a fixed minor and degenerate graphs. 1