Abstract. 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 vertex coloring 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 matrix estimation problems. The framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance-2 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 matrix as 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.

SUMMARYFinding a good graph coloring quickly is often a crucial phase in the development of efficient, parallel algorithms for many scientific and engineering applications. In this paper we consider the problem of solving the graph coloring problem itself in parallel. We present a simple and fast parallel graph coloring heuristic that is well suited for shared memory programming and yields an almost linear speedup on the PRAM model. We also present a second heuristic that improves on the number of colors used. The heuristics have been implemented using OpenMP. Experiments conducted on an SGI Cray Origin 2000 supercomputer using very large graphs from finite element methods and eigenvalue computations validate the theoretical run-time analysis.

Abstract. The maximum clique problem is a well known NP-Hard problem with applications in data mining, network analysis, information retrieval and many other areas related to the World Wide Web. There exist several algorithms for the problem with acceptable runtimes for certain classes of graphs, but many of them are infeasible for massive graphs. We present a new exact algorithm that employs novel pruning techniques and is able to quickly find maximum cliques in large sparse graphs. Extensive experiments on different kinds of synthetic and real-world graphs show that our new algorithm can be orders of magnitude faster than existing algorithms. We also present a heuristic that runs orders of magnitude faster than the exact algorithm while providing optimal or near-optimal solutions.

We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. The method exhibits a roughly linear runtime scaling over real-world networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Our method employs a branch and bound strategy with novel and aggressive pruning techniques. For instance, we use the core number of a vertex in combination with a good heuristic clique finder to efficiently remove the vast majority of the search space. In addition, we parallelize the exploration of the search tree. During the search, processes immediately communicate changes to upper and lower bounds on the size of maximum clique, which occasionally results in a super-linear speedup because vertices with large search spaces can be pruned by other processes. We apply the algorithm to two problems: to compute temporal strong components and to compress graphs.

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We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. Despite clique's status as an NP-hard problem with poor approximation guarantees, our method exhibits nearly linear runtime scaling over realworld networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Key to the efficiency of our algorithm are an initial heuristic procedure that finds a large clique quickly and a parallelized branch and bound strategy with aggressive pruning and ordering techniques. We use the algorithm to compute the largest temporal strong components of temporal contact networks.We propose a fast, parallel maximum clique finder wellsuited for applications involving large sparse graphs. Our algorithm is a branch and bound method with novel and aggressive pruning strategies. Branch and bound type algorithms for maximum clique explore all maximal cliques that cannot be pruned via search tree optimizations [3,7,5,8]. They differ chiefly in the way the pruning is done. Our algorithm is distinguished by several features. First, it begins by finding a large clique using a near linear-time heuristic; the obtained solution is checked for optimality before the algorithm proceeds any further, and the algorithm is terminated if the solution is found to be optimal. Second, we use this heuristic clique, in combination with (tight) upper bounds on the largest clique, to aggressively prune. The upper bounds are computed at the level of the input graph or individual neighborhoods. Third, we use implicit graph edits and periodic full graph updates in order to keep our

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BackgroundShort-sequence repeats (SSRs) occur in both prokaryotic and eukaryotic DNA, inter- and intragenically, and may be exact or inexact copies. When heterogeneous SSRs are present in a given locus, we can take advantage of the pattern of different repeats to genotype strains based on the SSRs. Cataloguing and tracking these repeats can be difficult as diverse groups of researchers are involved in the identification of the repeats. Additionally, the task is error-prone when done manually.ResultsWe developed RepeatAnalyzer, a new software tool capable of tracking, managing, analysing and cataloguing SSRs and genotypes using Anaplasma marginale as a model species. RepeatAnalyzer’s analysis capability includes novel metrics for measuring regional genetic diversity (corresponding to variety and regularity of SSR occurrence). As a part of its visualization capabilities, RepeatAnalyzer produces high quality maps of the geographic distribution of genotypes or SSRs over a region of interest. RepeatAnalyzer’s repeat identification functionality was validated for all SSRs and genotypes reported in 21 publications, using 380 A. marginale isolates gathered from the five publications within that list that provided access to their isolates. The tool produced accurate genotyping results in every case. In addition, it uncovered a number of errors in the published literature: 11 cases where SSRs were misreported, 5 cases where two different SSRs had been given the same name, and 16 cases where two or more names had been given to a single SSR. The analysis and visualization functionalities of the tool are demonstrated using several examples.ConclusionsRepeatAnalyzer is a robust software tool that can be used for storing, managing, and analysing short-sequence repeats for the purpose of strain identification. The tool can be used for any set of SSRs regardless of species. When applied to A. marginale, our test case, we show that genotype lengths for a given region follow a normal distribution, while SSR frequencies follow a power-law-like distribution. Further, we find that over 90 % of repeats are 28 to 29 amino acids long, which is in agreement with conventional wisdom. Lastly, our analysis reveals that the most common edit distance is five or six, which is counter-intuitive since we expected that result to be closer to one, resulting from the simplest change from one repeat to another.Electronic supplementary materialThe online version of this article (doi:10.1186/s12864-016-2686-2) contains supplementary material, which is available to authorized users.