Communities of vertices within a giant network such as the World-Wide-Web are likely to be vastly smaller than the network itself. However, Fortunato and Barthelemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than {radical} L/2 edges, where L is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barthelemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with fewer than {radical} W{epsilon}/2 total edge weight, where W is the total edge weight in the network and {epsilon} is the maximum weight of an inter-community edge. If {epsilon} is small, then small communities can be resolved. Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low {epsilon}, we modify the 'CNM' community detection algorithm to maximize weighted modularity, and show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed.
In this paper, we introduce EXACT, the EXperimental Algorithmics Computational Toolkit. EXACT is a software framework for describing, controlling, and analyzing computer experiments. It provides the experimentalist with convenient software tools to ease and organize the entire experimental process, including the description of factors and levels, the design of experiments, the control of experimental runs, the archiving of results, and analysis of results. As a case study for EXACT, we describe its interaction with FAST, the Sandia Framework for Agile Software Testing. EXACT and FAST now manage the nightly testing of several large software projects at Sandia. We also discuss EXACT's advanced features, which include a driver module that controls complex experiments such as comparisons of parallel algorithms. Copyright 2007 ACM.
Discrete models of large, complex systems like national infrastructures and complex logistics frameworks naturally incorporate many modeling uncertainties. Consequently, there is a clear need for optimization techniques that can robustly account for risks associated with modeling uncertainties. This report summarizes the progress of the Late-Start LDRD 'Robust Analysis of Largescale Combinatorial Applications'. This project developed new heuristics for solving robust optimization models, and developed new robust optimization models for describing uncertainty scenarios.
We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.
This report summarizes the research and development performed from October 2002 to September 2004 at Sandia National Laboratories under the Laboratory-Directed Research and Development (LDRD) project ''Massively-Parallel Linear Programming''. We developed a linear programming (LP) solver designed to use a large number of processors. LP is the optimization of a linear objective function subject to linear constraints. Companies and universities have expended huge efforts over decades to produce fast, stable serial LP solvers. Previous parallel codes run on shared-memory systems and have little or no distribution of the constraint matrix. We have seen no reports of general LP solver runs on large numbers of processors. Our parallel LP code is based on an efficient serial implementation of Mehrotra's interior-point predictor-corrector algorithm (PCx). The computational core of this algorithm is the assembly and solution of a sparse linear system. We have substantially rewritten the PCx code and based it on Trilinos, the parallel linear algebra library developed at Sandia. Our interior-point method can use either direct or iterative solvers for the linear system. To achieve a good parallel data distribution of the constraint matrix, we use a (pre-release) version of a hypergraph partitioner from the Zoltan partitioning library. We describe the design and implementation of our new LP solver called parPCx and give preliminary computational results. We summarize a number of issues related to efficient parallel solution of LPs with interior-point methods including data distribution, numerical stability, and solving the core linear system using both direct and iterative methods. We describe a number of applications of LP specific to US Department of Energy mission areas and we summarize our efforts to integrate parPCx (and parallel LP solvers in general) into Sandia's massively-parallel integer programming solver PICO (Parallel Interger and Combinatorial Optimizer). We conclude with directions for long-term future algorithmic research and for near-term development that could improve the performance of parPCx.
We consider the accuracy of predictions made by integer programming (IP) models of sensor placement for water security applications. We have recently shown that IP models can be used to find optimal sensor placements for a variety of different performance criteria (e.g. minimize health impacts and minimize time to detection). However, these models make a variety of simplifying assumptions that might bias the final solution. We show that our IP modeling assumptions are similar to models developed for other sensor placement methodologies, and thus IP models should give similar predictions. However, this discussion highlights that there are significant differences in how temporal effects are modeled for sensor placement. We describe how these modeling assumptions can impact sensor placements.
In recent years, several integer programming models have been proposed to place sensors in municipal water networks in order to detect intentional or accidental contamination. Although these initial models assumed that it is equally costly to place a sensor at any place in the network, there clearly are practical cost constraints that would impact a sensor placement decision. Such constraints include not only labor costs but also the general accessibility of a sensor placement location. In this paper, we extend our integer program to explicitly model the cost of sensor placement. We partition network locations into groups of varying placement cost, and we consider the public health impacts of contamination events under varying budget constraints. Thus our models permit cost/benefit analyses for differing sensor placement designs. As a control for our optimization experiments, we compare the set of sensor locations selected by the optimization models to a set of manually-selected sensor locations.
We present a model for optimizing the placement of sensors in municipal water networks to detect maliciously injected contaminants. An optimal sensor configuration minimizes the expected fraction of the population at risk. We formulate this problem as a mixed-integer program, which can be solved with generally available solvers. We find optimal sensor placements for three test networks with synthetic risk and population data. Our experiments illustrate that this formulation can be solved relatively quickly and that the predicted sensor configuration is relatively insensitive to uncertainties in the data used for prediction.
We give processor-allocation algorithms for grid architectures, where the objective is to select processors from a set of available processors to minimize the average number of communication hops. The associated clustering problem is as follows: Given n points in R{sup d}, find a size-k subset with minimum average pairwise L{sub 1} distance.We present a natural approximation algorithm and show that it is a 7/4-approximation for 2D grids. In d dimensions, the approximation guarantee is 2 - 1/2d, which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d and report on experimental results.
The Computational Plant or Cplant is a commodity-based distributed-memory supercomputer under development at Sandia National Laboratories. Distributed-memory supercomputers run many parallel programs simultaneously. Users submit their programs to a job queue. When a job is scheduled to run, it is assigned to a set of available processors. Job runtime depends not only on the number of processors but also on the particular set of processors assigned to it. Jobs should be allocated to localized clusters of processors to minimize communication costs and to avoid bandwidth contention caused by overlapping jobs. This report introduces new allocation strategies and performance metrics based on space-filling curves and one dimensional allocation strategies. These algorithms are general and simple. Preliminary simulations and Cplant experiments indicate that both space-filling curves and one-dimensional packing improve processor locality compared to the sorted free list strategy previously used on Cplant. These new allocation strategies are implemented in Release 2.0 of the Cplant System Software that was phased into the Cplant systems at Sandia by May 2002. Experimental results then demonstrated that the average number of communication hops between the processors allocated to a job strongly correlates with the job's completion time. This report also gives processor-allocation algorithms for minimizing the average number of communication hops between the assigned processors for grid architectures. The associated clustering problem is as follows: Given n points in {Re}d, find k points that minimize their average pairwise L{sub 1} distance. Exact and approximate algorithms are given for these optimization problems. One of these algorithms has been implemented on Cplant and will be included in Cplant System Software, Version 2.1, to be released. In more preliminary work, we suggest improvements to the scheduler separate from the allocator.
We present a model for optimizing the placement of sensors in municipal water networks to detect maliciously-injected contaminants. An optimal sensor configuration minimizes the expected fraction of the population at risk. We formulate this problem as an integer program, which can be solved with generally available IP solvers. We find optimal sensor placements for three real networks with synthetic risk and population data. Our experiments illustrate that this formulation can be solved relatively quickly, and that the predicted sensor configuration is relatively insensitive to uncertainties in the data used for prediction.
This report describes the design of PICO, a C++ framework for implementing general parallel branch-and-bound algorithms. The PICO framework provides a mechanism for the efficient implementation of a wide range of branch-and-bound methods on an equally wide range of parallel computing platforms. We first discuss the basic architecture of PICO, including the application class hierarchy and the package's serial and parallel layers. We next describe the design of the serial layer, and its central notion of manipulating subproblem states. Then, we discuss the design of the parallel layer, which includes flexible processor clustering and communication rates, various load balancing mechanisms, and a non-preemptive task scheduler running on each processor. We describe the application of the package to a branch-and-bound method for mixed integer programming, along with computational results on the ASCI Red massively parallel computer. Finally we describe the application of the branch-and-bound mixed-integer programming code to a resource constrained project scheduling problem for Pantex.
The Department of Energy (DOE) national laboratories have one of the longest and most consistent histories of supercomputer use. The authors summarize the architecture of DOE`s new supercomputers that are being built for the Accelerated Strategic Computing Initiative (ASCI). The authors then argue that in the near future scaled-down versions of these supercomputers with petaflop-per-weekend capabilities could become widely available to hundreds of research and engineering departments. The availability of such computational resources will allow simulation of physical phenomena to become a full-fledged third branch of scientific exploration, along with theory and experimentation. They describe the ASCI and other supercomputer applications at Sandia National Laboratories, and discuss which lessons learned from Sandia`s long history of supercomputing can be applied in this new setting.
This report describes the research performed under the laboratory-Directed Research and Development (LDRD) grant {open_quotes}A new approach to protein function and structure prediction{close_quotes}, funded FY94-6. We describe the goals of the research, motivate and list our improvements to the state of the art in multiple sequence alignment and phylogeny (evolutionary tree) construction, but leave technical details to the six publications resulting from this work. At least three algorithms for phylogeny construction or tree consensus have been implemented and used by researchers outside of Sandia.
A natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time. In this paper we consider the problem of scheduling n jobs that are released over time in order to minimize the average completion time of the set of jobs. In contrast to the problem of minimizing average completion time when all jobs are available at time 0, all the problems that we consider are NP-hard, and essentially nothing was known about constructing good approximations in polynomial time. We give the first constant-factor approximation algorithms for several variants of the single and parallel machine model. Many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both. Many of the algorithms generalize to the minimization of average weighted completion time as well.
Inferring phylogenetic trees is a fundamental problem in computational-biology. We present a new objective criterion, the phylogenetic number, for evaluating evolutionary trees for species defined by biomolecular sequences or other qualitative characters. The phylogenetic number of a tree T is the maximum number of times that any given character state arises in T. By contrast, the classical parsimonycriterion measures the total number of times that different character states arise in T. We consider the following related problems: finding the tree with minimum phylogenetic number, and computing the phylogenetic number of a given topology in which only the leaves are labeled by species. When the number of states is bounded (as is the case for biomolecular sequence characters), we can solve the second problem in polynomial time. We can also compute a fixed-topology 2-phylogeny (when one exists) for an arbitrary number of states. This algorithm can be used to further distinguish trees that are equal under parsimony. We also consider a number of other related problems.