The cyclic coloring problem and estimation of spare hessian matrices
SIAM Journal on Algebraic and Discrete Methods
Algorithms for Square Roots of Graphs
SIAM Journal on Discrete Mathematics
Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
ACM Transactions on Mathematical Software (TOMS)
Graph classes: a survey
An Interior-Point Algorithm for Nonconvex Nonlinear Programming
Computational Optimization and Applications - Special issue on computational optimization—a tribute to Olvi Mangasarian, part II
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Mathematical Programming: Series A and B
Computing sparse Hessians with automatic differentiation
ACM Transactions on Mathematical Software (TOMS)
New Acyclic and Star Coloring Algorithms with Application to Computing Hessians
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Acyclic colorings of graph subdivisions
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Acyclically 3-colorable planar graphs
Journal of Combinatorial Optimization
Acyclic colorings of graph subdivisions revisited
Journal of Discrete Algorithms
A polyhedral study of the acyclic coloring problem
Discrete Applied Mathematics
Computing the sparsity pattern of Hessians using automatic differentiation
ACM Transactions on Mathematical Software (TOMS)
ColPack: Software for graph coloring and related problems in scientific computing
ACM Transactions on Mathematical Software (TOMS)
Acyclic coloring with few division vertices
Journal of Discrete Algorithms
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The computation of a sparse Hessian matrix H using automatic differentiation (AD) can be made efficient using the following four-step procedure: (1) Determine the sparsity structure of H, (2) obtain a seed matrix S that defines a column partition of H using a specialized coloring on the adjacency graph of H, (3) compute the compressed Hessian matrix B ≡ HS, and (4) recover the numerical values of the entries of H from B. The coloring variant used in the second step depends on whether the recovery in the fourth step is direct or indirect: a direct method uses star coloring and an indirect method uses acyclic coloring. In an earlier work, we had designed and implemented effective heuristic algorithms for these two NP-hard coloring problems. Recently, we integrated part of the developed software with the AD tool ADOL-C, which has recently acquired a sparsity detection capability. In this paper, we provide a detailed description and analysis of the recovery algorithms and experimentally demonstrate the efficacy of the coloring techniques in the overall process of computing the Hessian of a given function using ADOL-C as an example of an AD tool. We also present new analytical results on star and acyclic coloring of chordal graphs. The experimental results show that sparsity exploitation via coloring yields enormous savings in runtime and makes the computation of Hessians of very large size feasible. The results also show that evaluating a Hessian via an indirect method is often faster than a direct evaluation. This speedup is achieved without compromising numerical accuracy.