This work aims at bringing advances in discrete optimization algorithms to solving practical engineering problems at scale. Often times, in many engineering design problems, there is a need to select a small set of influential or representative elements from a large ground set of entities in an optimal fashion. Submodular optimization provides for a formal way to solve such problems. Common examples with infrastructure systems involve sensor placement and identification of key entities with certain objectives. However, scaling these approaches to large infrastructure systems can be challenging because of the high computational complexity of the overall framework that include the optimization algorithms as well as high-complexity compute-oracles that provide the necessary objective function values. In this work, we explore a well-studied and widely-applicable paradigm, namely leader-selection in a multi-agent networked setting in the context of scalable methodologies. We demonstrate novel frameworks that utilize variations of accelerated submodular optimization algorithms along with linear-algebraic methods that can help accelerate the oracle computations. We further explore this combination in conjunction with graph partitioning paradigms to take advantage of the accelerated algorithms in a distributed setting. Finally we demonstrate the key findings on a practical problem in an operational setting. For this, we leverage an example road network with approximately 18k nodes and 27k edges in a traffic control application, where we seek a limited number of k = 200 key intersections. This problem can be solved in a serial setting in just under 5 hours providing more than 2 orders of magnitude speed-up over methods that do not consider acceleration techniques.

Subgraph isomorphism is a pattern-matching algorithm widely used in many domains such as chem-informatics, bioinformatics, databases, and social network analysis. It is computationally expensive and is a proven NP-hard problem. The massive parallelism in GPUs is well suited for solving subgraph isomorphism. However, current GPU implementations are far from the achievable performance. Moreover, the enormous memory requirement of current approaches limits the problem size that can be handled. This work analyzes the fundamental challenges associated with processing subgraph isomorphism on GPUs and develops an efficient GPU implementation. We also develop a GPU-friendly trie-based data structure to drastically reduce the intermediate storage space requirement, enabling large benchmarks to be processed. We also develop the first distributed sub-graph isomorphism algorithm for GPUs. Our experimental evaluation demonstrates the efficacy of our approach by comparing the execution time and number of cases that can be handled against the state-of-the-art GPU implementations.

We consider an assignment problem arising in Fermionic-swap based mapping of the one-body and two-body interaction terms in simulating time evolution of a sparse second-quantized electronic structure Hamiltonian on a quantum computer. Relative efficiency of different assignment algorithms depends on the relative costs of performing a swap and computing a Hamiltonian interaction term. Under the assumption that the interaction term cost dominates the computation, we develop an iterative algorithm that uses minimum cost linear assignment (MinLA) and matching for one-body interactions, and hypergraph optimal linear arrangement (HOLA) and partial distance-2 coloring for two-body interactions, to exploit arbitrary sparsity in the Hamiltonian for efficient computation. Using a set of 122 problems from computational chemistry, we demonstrate performance improvements up to 100% relative to the state-of-the-art approach for one-body terms and up to 86% utilization for two-body terms relative to a theoretical peak utilization. To the best of our knowledge, this is the first study to exploit arbitrary sparsity in orbital interactions for efficient computation on one-dimensional qubit connectivity layouts. The proposed algorithms lay a foundation for extension to map general k-body interactions that arise in many domains onto generalized qubit connectivity layouts available in current and future quantum systems.

Traditional implementations of parallel graph operations on distributed memory platforms are written using Message Passing Interface (MPI) point-to-point communication primitives such as Send-Recv (blocking and nonblocking). Apart from this classical model, the MPI community has over the years added other communication models; however, their suitability for handling the irregular traffic workloads typical of graph operations remain comparatively less explored. Our aim in this paper is to study these relatively underutilized communication models of MPI for graph applications. More specifically, we evaluate MPI's one-sided programming, or Remote Memory Access (RMA), and nearest neighborhood collectives using a process graph topology. There are features in these newer models that are intended to better map to irregular communication patterns, as exemplified in graph algorithms. As a concrete application for our case study, we use distributed memory implementations of an approximate weighted graph matching algorithm to investigate performances of MPI3 RMA and neighborhood collective operations compared to nonblocking Send-Recv. A matching in a graph is a subset of edges such that no two matched edges are incident on the same vertex. A maximum weight matching is a matching of maximum weight computed as the sum of the weights of matched edges. Execution of graph matching is dominated by high volume of irregular memory accesses, making it an ideal candidate for studying the effects of various MPI communication models on graph applications at scale. Our neighborhood collectives and RMA implementations yield up to 6× speedup over traditional nonblocking Send-Recv implementations on thousands of cores of the NERSC Cori supercomputer. We believe the lessons learned from this study can be adopted to benefit a wider range of graph applications.

We describe a paradigm for designing parallel algorithms via approximation, and illustrate it on the b-Edge Cover problem. A b-Edge Cover of minimum weight in a graph is a subset C of its edges such that at least a specified number b(v) of edges in C is incident on each vertex v, and the sum of the edge weights in C is minimum. The Greedy algorithm and a variant, the LSE algorithm, provide 3/2-approximation guarantees in the worst-case for this problem, but these algorithms have limited parallelism. Hence we design two new 2-approximation algorithms with greater concurrency. The MCE algorithm reduces the computation of a b-Edge Cover to that of finding a b'-Matching, by exploiting the relationship between these subgraphs in an approximation context. The LSENW is derived from the LSE algorithm using static edge weights rather than dynamically computing effective edge weights. This relaxation gives S-LSE a worse approximation guarantee but makes it more amenable to parallelization. We prove that both the MCE and S-LSE algorithms compute the same b-EDGE COVER with at most twice the weight of the minimum weight edge cover. In practice, the 2-approximation and 3/2-approximation algorithms compute edge covers of weight within 10% the optimal. We implement three of the approximation algorithms, MCE, LSE, and S-LSE, on shared memory multi-core machines, including an Intel Xeon and an IBM Power8 machine with 8 TB memory. The MCE algorithm is the fastest of these by an order of magnitude or more. It computes an edge cover in a graph with billions of edges in 20 seconds using two hundred threads on the IBM Power8. We also show that the parallel depth and work can be bounded for the Suitor and b-Suitor algorithms when edge weights are random.

In most real-world networks, the nodes/vertices tend to be organized into tightly-knit modules known as communities or clusters, such that nodes within a community are more likely to be "related" to one another than they are to the rest of the network. The goodness of partitioning into communities is typically measured using a well known measure called modularity. However, modularity optimization is an NP-complete problem. In 2008, Blondel, et al. introduced a multi-phase, iterative heuristic for modularity optimization, called the Louvain method. Owing to its speed and ability to yield high quality communities, the Louvain method continues to be one of the most widely used tools for serial community detection. In this paper, we present the design of a distributed memory implementation of the Louvain algorithm for parallel community detection. Our approach begins with an arbitrarily partitioned distributed graph input, and employs several heuristics to speedup the computation of the different steps of the Louvain algorithm. We evaluate our implementation and its different variants using real-world networks from various application domains (including internet, biology, social networks). Our MPI+OpenMP implementation yields about 7x speedup (on 4K processes) for soc-friendster network (1.8B edges) over a state-of-the-art shared memory multicore implementation (on 64 threads), without compromising output quality. Furthermore, our distributed implementation was able to process a larger graph (uk-2007; 3.3B edges) in 32 seconds on 1K cores (64 nodes) of NERSC Cori, when the state-of-the-art shared memory implementation failed to run due to insufficient memory on a single Cori node containing 128 GB of memory.

A b-MATCHING is a subset of edges M such that at most b(v) edges in M are incident on each vertex v, where b(v) is specified. We present a distributed-memory parallel algorithm, b-SUITOR, that computes a b-MATCHING with more than half the maximum weight in a graph with weights on the edges. The approximation algorithm is designed to have high concurrency and low time complexity. We organize the implementation of the algorithm in terms of asynchronous supersteps that combine computation and communication, and balance the computational work and frequency of communication to obtain high performance. Since the performance of the b-SUITOR algorithm is strongly influenced by communication, we present several strategies to reduce the communication volume. We implement the algorithm using a hybrid strategy where inter-node communication uses MPI and intra-node computation is done with OpenMP threads. We demonstrate strong and weak scaling of b-SUITOR up to 16K processors on two supercomputers at NERSC. We compute a b-MATCHING in a graph with 2 billion edges in under 4 seconds using 16K processors.

We design, implement, and evaluate algorithms for computing a matching of maximum cardinality in a bipartite graph on multicore and massively multithreaded computers. As computers with larger numbers of slower cores dominate the commodity processor market, the design of multithreaded algorithms to solve large matching problems becomes a necessity. Recent work on serial algorithms for the matching problem has shown that their performance is sensitive to the order in which the vertices are processed for matching. In a multithreaded environment, imposing a serial order in which vertices are considered for matching would lead to loss of concurrency and performance. But this raises the question: {\em Would parallel matching algorithms on multithreaded machines improve performance over a serial algorithm?}We answer this question in the affirmative. We report efficient multithreaded implementations of three classes of algorithms based on their manner of searching for augmenting paths: breadth-first-search, depth-first-search, and a combination of both. The Karp-Sipser initialization algorithm is used to make the parallel algorithms practical. We report extensive results and insights using three shared-memory platforms (a 48-core AMD Opteron, a 32-coreIntel Nehalem, and a 128-processor Cray XMT) on a representative set of real-world and synthetic graphs. To the best of our knowledge, this is the first study of augmentation-based parallel algorithms for bipartite cardinality matching that demonstrates good speedups on multithreaded shared memory multiprocessors.

Network alignment is an optimization problem to find the best one-to-one map between the vertices of a pair of graphs that overlaps as many edges as possible. It is a relaxation of the graph isomorphism problem and is closely related to the subgraph isomorphism problem. The best current approaches are entirely heuristic and iterative in nature. They generate real-valued heuristic weights that must be rounded to find integer solutions. This rounding requires solving a bipartite maximum weight matching problem at each iteration in order to avoid missing high quality solutions. We investigate substituting a parallel, half-approximation for maximum weight matching instead of an exact computation. Our experiments show that the resulting difference in solution quality is negligible. We demonstrate almost a 20-fold speedup using 40 threads on an 8 processor Intel Xeon E7-8870 system and now solve real-world problems in 36 seconds instead of 10 minutes.