An official website of the United States government
We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted $$b$$ -matching, and minimization versions of weighted edge cover and $$b$$ -edge cover. Exact algorithms for these problems are impractical for massive graphs with several millions of edges. For each problem we discuss theoretical foundations, the design of several linear or near-linear time approximation algorithms, their implementations on serial and parallel computers, and applications. Our focus is on practical algorithms that yield good performance on modern computer architectures with multiple threads and interconnected processors. We also include information about the software available for these problems.
more »
« less
A Parallel Approximation Algorithm for Maximizing Submodular b-Matching
We design new serial and parallel approximation algorithms for computing a maximum weight b-matching in an edge-weighted graph with a submodular objective function. This problem is NP-hard; the new algorithms have approximation ratio 1/3, and are relaxations of the Greedy algorithm that rely only on local information in the graph, making them parallelizable. We have designed and implemented Local Lazy Greedy algorithms for both serial and parallel computers. We have applied the approximate submodular b-matching algorithm to assign tasks to processors in the computation of Fock matrices in quantum chemistry on parallel computers. The assignment seeks to reduce the run time by balancing the computational load on the processors and bounding the number of messages that each processor sends. We show that the new assignment of tasks to processors provides a four fold speedup over the currently used assignment in the NWChemEx software on 8000 processors on the Summit supercomputer at Oak Ridge National Lab.
more »
« less
- Award ID(s):
- 1637534
- PAR ID:
- 10300888
- Editor(s):
- Bender, M.; Gilbert, J.; Hendrickson, B.; Sullivan, B.
- Date Published:
- Journal Name:
- Proceedings of the 2021 SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)
- Volume:
- 1
- Issue:
- 1
- Page Range / eLocation ID:
- 45-56
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We describe a 3/2-approximation algorithm, \lse, for computing a b-edgecover of minimum weight in a graph with weights on the edges. The b-edgecover problem is a generalization of the better-known Edge Cover problem in graphs, where the objective is to choose a subset C of edges in the graph such that at least a specified number b(v) of edges in C are incident on each vertex v. In the weighted b-edgecover problem, we minimize the sum of the weights of the edges in C. We prove that the Locally Subdominant edge (LSE) algorithm computes the same b-edge cover as the one obtained by the Greedy algorithm for the problem. However, the Greedy algorithm requires edges to be sorted by their effective weights, and these weights need to be updated after each iteration. These requirements make the Greedy algorithm sequential and impractical for massive graphs. The LSE algorithm avoids the sorting step, and is amenable for parallelization. We implement the algorithm on a serial machine and compare its performance against a collection of approximation algorithms for the b-edge cover problem. Our results show that the algorithm is 3 to 5 times faster than the Greedy algorithm on a serial processor. The approximate edge covers obtained by the LSE algorithm have weights greater by at most 17% of the optimal weight for problems where we could compute the latter. We also investigate the relationship between the b-edge cover and the b-matching problems, show that the latter has a faster implementation since edge weights are static in this algorithm, and obtain a heuristic solution for the former from the latter.more » « less
-
Mutzel, Petra; Prezza, Nicola (Ed.)We describe a parallel approximation algorithm for maximizing monotone submodular functions subject to hereditary constraints on distributed memory multiprocessors. Our work is motivated by the need to solve submodular optimization problems on massive data sets, for practical contexts such as data summarization, machine learning, and graph sparsification. Our work builds on the randomized distributed RandGreeDI algorithm, proposed by Barbosa, Ene, Nguyen, and Ward (2015). This algorithm computes a distributed solution by randomly partitioning the data among all the processors and then employing a single accumulation step in which all processors send their partial solutions to one processor. However, for large problems, the accumulation step exceeds the memory available on a processor, and the processor which performs the accumulation becomes a computational bottleneck. Hence we propose a generalization of the RandGreeDI algorithm that employs multiple accumulation steps to reduce the memory required. We analyze the approximation ratio and the time complexity of the algorithm (in the BSP model). We evaluate the new GreedyML algorithm on three classes of problems, and report results from large-scale data sets with millions of elements. The results show that the GreedyML algorithm can solve problems where the sequential Greedy and distributed RandGreeDI algorithms fail due to memory constraints. For certain computationally intensive problems, the GreedyML algorithm is faster than the RandGreeDI algorithm. The observed approximation quality of the solutions computed by the GreedyML algorithm closely matches those obtained by the RandGreeDI algorithm on these problems.more » « less
-
We describe a paradigm for designing parallel algorithms via approximation, and illustrate it on the b-edgecover problem. A b-edgecover 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-edgecover to that of finding a b'-matching, by exploiting the relationship between these subgraphs in an approximation context. The LSE-NW is derived from the LSEalgorithm using static edge weights rather than dynamically computing effective edge weights. This relaxation gives LSE a worse approximation guarantee but makes it more amenable to parallelization. We prove that both the MCE and LSE-NW algorithms compute the same b-edgecover 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 LSE-NW 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.more » « less
-
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, \bsuitor, 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 super-steps 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 16,000 processors on two supercomputers at NERSC. We compute a b-matching in a graph with 2 billion edges in under 4 seconds using 16,000 processors.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611976830.5
