We consider a hypergraph model for the protein complex network obtained from a large-scale experimental study to characterize the proteome of the yeast. Our model views the yeast proteome as a hypergraph, with the proteins corresponding to vertices and the complexes corresponding to hyperedges. Previous work has modeled the protein complex data as a protein-protein interaction graph or as a complex intersection graph; both models lose information and require more space. Our results show that the yeast protein complex hyper-graph is a small-world and power-law hypergraph. We design an algorithm for computing the -core of a hypergraph, and use it to identify the core proteome, the maximum core of the protein complex hypergraph. We show that the core proteome of the yeast is enriched in essential and homologous proteins. We implement greedy approximation algorithms for variant minimum weight vertex covers of a hypergraph; these algorithms can be used to improve the reliability and efficiency of the experimental method that identifies the protein complex network.

T he 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.

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