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The classical Birkhoff–von Neumann (BvN) decomposition expresses a given doubly stochastic matrix as a convex combination of permutation matrices. We investigate the BvN decomposition of ...

We investigate the performance of algorithms for sparse tensor-sparse tensor multiplication (SpGETT). This operation, also called sparse tensor contraction, is a higher order analogue of the sparse matrix-sparse matrix multiplication (SpGEMM) operation. Therefore, SpGETT can be performed by first converting the input tensors into matrices, then invoking high performance variants of SpGEMM, and finally reconverting the resultant matrix into a tensor. Alternatively, one can carry out the scalar operations underlying SpGETT in the realm of tensors without matrix formulation. We discuss the building blocks in both approaches and formulate a hashing-based method to avoid costly search or redirection operations. We present performance results with the current state-of-the-art SpGEMM-based approaches, existing SpGETT approaches, and a carefully implemented SpGETT approach with a new fine-tuned hashing method, proposed in this article. We evaluate the methods on real world tensors by contracting a tensor with itself along varying dimensions. Our proposed hashing-based method for SpGETT consistently outperforms the state-of-the-art method, achieving a 25% reduction in sequential execution time on average and a 21% reduction in parallel execution time on average across a variety of input instances.

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition.

We consider the problem of querying the existence of hyperedges in hypergraphs. More formally, given a hypergraph, we need to answer queries of the form: “Does the following set of vertices form a hyperedge in the given hypergraph?” Our aim is to set up data structures based on hashing to answer these queries as fast as possible. We propose an adaptation of a well-known perfect hashing approach for the problem at hand. We analyze the space and runtime complexity of the proposed approach and experimentally compare it with the state-of-the-art hashing-based solutions. Experiments demonstrate the efficiency of the proposed approach with respect to the state-of-the-art.

When scheduling a directed acyclic graph (DAG) of tasks with communication costs on computational platforms, a good trade-off between load balance and data locality is necessary. List-based scheduling techniques are commonly-used greedy approaches for this problem. The downside of list-scheduling heuristics is that they are incapable of making short-term sacrifices for the global efficiency of the schedule. In this work, we describe new list-based scheduling heuristics based on clustering for homogeneous platforms, under the realistic duplex single-port communication model. Our approach uses an acyclic partitioner for DAGs for clustering. The clustering enhances the data locality of the scheduler with a global view of the graph. Furthermore, since the partition is acyclic, we can schedule each part completely once its input tasks are ready to be executed. We present an extensive experimental evaluation showing the trade-offs between the granularity of clustering and the parallelism, and how this affects the scheduling. Furthermore, we compare our heuristics to the best state-of-the-art list-scheduling and clustering heuristics, and obtain more than three times better makespan in cases with many communications.