Graph Coloring in Optimization Revisited

We revisit the role of graph coloring in modeling a variety of matrix partitioning problems that arise in numerical determination of large sparse Jacobian and Hessian matrices. The problems considered in this paper correspond to the various scenarios under which a matrix computation, or estimation, may be carried out, i.e., the particular problem depends on whether the matrix to be computed is symmetric or nonsymmetric, whether a one-dimensional or a two-dimensional partition is to be used, whether a direct or a substitution based evaluation scheme is to be employed, and whether all nonzero entries of the matrix or only a subset need to be computed. The resulting complex partitioning problems are studied within a unified graph theoretic framework where each problem is formulated as a variant of a coloring problem. Our study integrates existing coloring formulations with new ones. As far as we know, the estimation of a subset of the nonzero entries of a matrix is investigated for the first time. The insight gained from the unified graph theoretic treatment is used to develop and analyze several new heuristic algorithms.

[1]  G. Nemhauser,et al.  Integer Programming , 2020 .

[2]  Uwe Naumann,et al.  Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph , 2004, Math. Program..

[3]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[4]  Alex Pothen,et al.  Parallel Distance-k Coloring Algorithms for Numerical Optimization , 2002, Euro-Par.

[5]  Uwe Naumann,et al.  Cheaper Jacobians by Simulated Annealing , 2002, SIAM J. Optim..

[6]  Madhav V. Marathe,et al.  Approximation Algorithms for Channel Assignment in Radio Networks , 1998 .

[7]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[8]  Andreas Griewank,et al.  Automatic Differentiation of Algorithms: From Simulation to Optimization , 2000, Springer New York.

[9]  T. Steihaug,et al.  Reducing the number of AD passes for computing a sparse Jacobian matrix , 2000 .

[10]  Assefaw Hadish Gebremedhin,et al.  Scalable parallel graph coloring algorithms , 2000, Concurr. Pract. Exp..

[11]  Magnús M. Halldórsson,et al.  Coloring powers of planar graphs , 2000, SODA '00.

[12]  Thomas F. Coleman,et al.  The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation , 1998, SIAM J. Sci. Comput..

[13]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[14]  Rajeev Motwani,et al.  Average-case analysis of algorithms for matchings and related problems , 1994, JACM.

[15]  Steven Skiena,et al.  Algorithms for Square Roots of Graphs , 1991, SIAM J. Discret. Math..

[16]  Wayne Goddard,et al.  Acyclic colorings of planar graphs , 1991, Discret. Math..

[17]  T. Coleman,et al.  The cyclic coloring problem and estimation of spare hessian matrices , 1986 .

[18]  Thomas F. Coleman,et al.  Software for estimating sparse Hessian matrices , 1985, TOMS.

[19]  P. Toint,et al.  Optimal estimation of Jacobian and Hessian matrices that arise in finite difference calculations , 1984 .

[20]  Thomas F. Coleman,et al.  Software for estimating sparse Jacobian matrices , 1984, ACM Trans. Math. Softw..

[21]  John D. Ramsdell,et al.  Estimation of Sparse Jacobian Matrices , 1983 .

[22]  S. Thomas McCormick,et al.  Optimal approximation of sparse hessians and its equivalence to a graph coloring problem , 1983, Math. Program..

[23]  J. J. Moré,et al.  Estimation of sparse jacobian matrices and graph coloring problems , 1983 .

[24]  Thomas F. Coleman,et al.  Estimation of sparse hessian matrices and graph coloring problems , 1982, Math. Program..

[25]  M. Powell,et al.  On the Estimation of Sparse Hessian Matrices , 1979 .

[26]  M. Halldórsson,et al.  On Powers of Chordal Graphs And Their Colorings , 2000 .

[27]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[28]  S. Hossain.  On the computation of sparse Jacobian matrices and Newton steps , 1998 .

[29]  T. Steihaug,et al.  Computing a sparse Jacobian matrix by rows and columns , 1998 .

[30]  A. Griewank,et al.  Automatic differentiation of algorithms : theory, implementation, and application , 1994 .

[31]  Owe Axelsson,et al.  On a Graphical Package for Nonlinear Partial Differential Equation Problems , 1977, IFIP Congress.