Oded Schwartz

Minimizing Communication in Linear Algebra

In 1981 Hong and Kung proved a lower bound on the amount of communication (amount of data moved between a small, fast memory and large, slow memory) needed to perform dense, n-by-n matrix multiplication using the conventional O(n3) algorithm, where the input matrices were too large to fit in the small, fast memory. In 2004 Irony, Toledo, and Tiskin gave a new proof of this result and extended it to the parallel case (where communication means the amount of data moved between processors). In both cases the lower bound may be expressed as Ω(#arithmetic_operations/M), where M is the size of the fast memory (or local memory in the parallel case). Here we generalize these results to a much wider variety of algorithms, including LU factorization, Cholesky factorization, LDLT factorization, QR factorization, the Gram–Schmidt algorithm, and algorithms for eigenvalues and singular values, i.e., essentially all direct methods of linear algebra. The proof works for dense or sparse matrices and for sequential or para...

On the complexity of approximating k -set packing

Abstract.Given a k-uniform hypergraph, the Maximumk -Set Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of

Communication optimal parallel multiplication of sparse random matrices

\Omega {\left( {k/\ln k} \right)}

Communication lower bounds and optimal algorithms for numerical linear algebra

unless P = NP. This improves the previous hardness of approximation factor of

On the Complexity of Approximating k-Dimensional Matching

k/2^{{O({\sqrt {\ln k} })}}

On the Complexity of Approximating TSP with Neighborhoods and Related Problems

by Trevisan. This result extends to the problem of k-Dimensional-Matching.