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24
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.
The L(h, k)Labelling Problem: A Survey and Annotated Bibliography
, 2006
"... Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at l ..."
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Cited by 29 (3 self)
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Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)labelling with minimum span. The L(h, k)labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach.
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.
Coloring squares of planar graphs with girth six
 SUBMITTED TO EUROPEAN JOURNAL OF COMBINATORICS
"... Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off ..."
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Cited by 19 (4 self)
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Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is ( ∆ + 2)colorable.
Graph Labellings with Variable Weights, a Survey
, 2007
"... Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings ..."
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Cited by 12 (1 self)
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Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λgraphs. We survey recent results obtained in this area. The concept of real number graph labellings was introduced a few years ago, and in the sequel, a more general concept of λgraphs appeared. Though the two concepts are quite new, they are so natural that there are already many results on each. In fact, even some older results fall in this area, but their authors used a different mathematical language to state their achievements. Since many of these results are so recent that they are just appearing in various journals, we would like to offer the reader a single reference for the state of art as well as to draw attention to some older results that fall in this area.
Data Networks
 Upper Saddle River
, 1992
"... filtering + clustering technique for powerlaw ..."
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DISTRIBUTEDMEMORY PARALLEL ALGORITHMS FOR DISTANCE2 COLORING AND THEIR APPLICATION TO DERIVATIVE COMPUTATION
, 2010
"... The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. ..."
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Cited by 8 (6 self)
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The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. We present efficient, distributedmemory, parallel heuristic algorithms for this NPhard problem as well as for two related problems used in the computation of Jacobians and Hessians. Parallel speedup is achieved through graph partitioning, speculative (iterative) coloring, and a BSPlike organization of parallel computation. Results from experiments conducted on a PC cluster employing up to 96 processors and using largesize realworld as well as synthetically generated test graphs show that the algorithms are scalable. In terms of quality of solution, the algorithms perform remarkably well—the number of colors used by the parallel algorithms was observed to be very close to the number used by the sequential counterparts, which in turn are quite often near optimal. Moreover, the experimental results show that the parallel distance2 coloring algorithm compares favorably with the alternative approach of solving the distance2 coloring problem on a graph G by first constructing the square graph G² and then applying a parallel distance1 coloring algorithm on G2. Implementations of the algorithms are made available via the Zoltan loadbalancing library.
Colourings of the Cartesian product of graphs and multiplicative Sidon sets
, 2005
"... Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choic ..."
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Cited by 8 (3 self)
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Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choices of F, several wellknown types of colourings fit into this framework, including acyclic colourings, star colourings, and distance2 colourings. This paper studies Ffree colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1, G2,..., Gd. Our main result establishes an upper bound on the Ffree chromatic number of H in terms of the maximum Ffree chromatic number of the Gi and the following numbertheoretic concept. A set S of natural numbers is kmultiplicative Sidon if ax = by implies a = b and x = y whenever x,y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a kmultiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k·max S. We then prove that the maximum density of a kmultiplicative Sidon set is Θ(1/log k). It follows that χ(H, F) ≤ O(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.
Strong colorings of hypergraphs
"... Abstract. A strong vertex coloring of a hypergraph assigns distinct colors to vertices that are contained in a common hyperedge. This captures many previously studied graph coloring problems. We present nearly tight upper and lower bound on approximating general hypergraphs, both offline and online. ..."
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Cited by 7 (0 self)
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Abstract. A strong vertex coloring of a hypergraph assigns distinct colors to vertices that are contained in a common hyperedge. This captures many previously studied graph coloring problems. We present nearly tight upper and lower bound on approximating general hypergraphs, both offline and online. We then consider various parameters that make coloring easier, and give a unified treatment. In particular, we give an algebraic scheme using integer programming to color graphs of bounded compositionwidth.