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23
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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 41 (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.
The efficient computation of sparse Jacobian matrices using automatic differentiation
 SIAM J. Sci. Comput
, 1996
"... This paper is concerned with the efficient computation of sparse Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of a graph coloring technique, bicoloring, to exploit the sparsity of the Jacobian matrix J and thereby allow for the ef ..."
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Cited by 30 (7 self)
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This paper is concerned with the efficient computation of sparse Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of a graph coloring technique, bicoloring, to exploit the sparsity of the Jacobian matrix J and thereby allow for the efficient determination of J using AD software. We analyze both a direct scheme and a substitution process. We discuss the results of numerical experiments indicating significant practical potential of this approach.
Reducing the Number of AD Passes for Computing a Sparse Jacobian Matrix
, 2000
"... A reduction in the computational work is possible if we do not require that the nonzeros of a Jacobian matrix be determined directly. If a column or row partition is available, the proposed substitution technique can be used to reduce the number of groups in the partition further. In this chapte ..."
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Cited by 11 (3 self)
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A reduction in the computational work is possible if we do not require that the nonzeros of a Jacobian matrix be determined directly. If a column or row partition is available, the proposed substitution technique can be used to reduce the number of groups in the partition further. In this chapter, we present a substitution method to determine the structure of sparse Jacobian matrices efficiently using forward, reverse, or a combination of forward and reverse modes of AD. Specifically, if it is true that the difference between the maximum number of nonzeros in a column or row and the number of groups in the corresponding partition is large, then the proposed method can save many AD passes. This assertion is supported by numerical examples.
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 11 (7 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.
Structured Automatic Differentiation
, 1998
"... eme which combines the forward and reverse modes of AD. Problem structure can be viewed in many di#erent ways; one way is to look at the granularity of the operations involved. For example, di#erentiation carried out at the matrixvector operations can lead to great savings in the time as well as sp ..."
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Cited by 9 (0 self)
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eme which combines the forward and reverse modes of AD. Problem structure can be viewed in many di#erent ways; one way is to look at the granularity of the operations involved. For example, di#erentiation carried out at the matrixvector operations can lead to great savings in the time as well as space requirements. Figuring out the kind of computation is another way to view structure, e.g., partially separable or composite functions whose structure can be exploited to get performance gains. In this thesis we develop a general structure framework which can be viewed hierarchically and allows for structure exploitation at various levels. For example, for time integration schemes employing stencils it is possible to exploit structure at both the stencil level and the timestep level. We also present some advanced structure exploitation ideas, e.g., parallelism in structured computations and using structure in implicit computations. The use of AD as a derivative computing e
APPROXIMATING MAXIMUM STABLE SET AND MINIMUM GRAPH COLORING PROBLEMS WITH THE POSITIVE SEMIDEFINITE RELAXATION
"... We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in m ..."
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Cited by 9 (1 self)
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We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.
Combinatorial scientific computing: The enabling power of discrete algorithms in computational science
 In 7th Intl. Mtg. High Perf. Comput. for Computational Sci. (VECPAR’06), Lecture Notes in Computer Science
, 2006
"... Abstract. Combinatorial algorithms have long played a crucial, albeit underrecognized role in scientific computing. This impact ranges well beyond the familiar applications of graph algorithms in sparse matrices to include mesh generation, optimization, computational biology and chemistry, data ana ..."
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Cited by 8 (1 self)
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Abstract. Combinatorial algorithms have long played a crucial, albeit underrecognized role in scientific computing. This impact ranges well beyond the familiar applications of graph algorithms in sparse matrices to include mesh generation, optimization, computational biology and chemistry, data analysis and parallelization. Trends in science and in computing suggest strongly that the importance of discrete algorithms in computational science will continue to grow. This paper reviews some of these many past successes and highlights emerging areas of promise and opportunity. 1
SecondOrder Information in Data Assimilation
, 2002
"... In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a firstorder optimality system. However, existence and ..."
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Cited by 8 (2 self)
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In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a firstorder optimality system. However, existence and uniqueness of the VDA problem along with convergence of the algorithms for its implementation depend on the convexity of the cost function. Properties of local convexity can be deduced by studying the Hessian of the cost function in the vicinity of the optimum. This shows the necessity of secondorder information to ensure a unique solution to the VDA problem.
Graph Coloring in Optimization Revisited
, 2002
"... We revisit the role of graph coloring in modeling problems that arise in efficient estimation of large sparse Jacobian and Hessian matrices using both finite difference (FD) and automatic differentiation (AD) techniques, in each case via direct methods. For Jacobian estimation using column partit ..."
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Cited by 6 (1 self)
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We revisit the role of graph coloring in modeling problems that arise in efficient estimation of large sparse Jacobian and Hessian matrices using both finite difference (FD) and automatic differentiation (AD) techniques, in each case via direct methods. For Jacobian estimation using column partitioning, we propose a new coloring formulation based on a bipartite graph representation. This is compared