5 research outputs found
Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving
Graph sparsification underlies a large number of algorithms, ranging from
approximation algorithms for cut problems to solvers for linear systems in the
graph Laplacian. In its strongest form, "spectral sparsification" reduces the
number of edges to near-linear in the number of nodes, while approximately
preserving the cut and spectral structure of the graph. In this work we
demonstrate a polynomial quantum speedup for spectral sparsification and many
of its applications. In particular, we give a quantum algorithm that, given a
weighted graph with nodes and edges, outputs a classical description of
an -spectral sparsifier in sublinear time
. This contrasts with the optimal classical
complexity . We also prove that our quantum algorithm is optimal
up to polylog-factors. The algorithm builds on a string of existing results on
sparsification, graph spanners, quantum algorithms for shortest paths, and
efficient constructions for -wise independent random strings. Our algorithm
implies a quantum speedup for solving Laplacian systems and for approximating a
range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will
appear in FOCS'20; v3: corrected a minor mistake in Appendix
Quantum and Classical Algorithms for Approximate Submodular Function Minimization
22 pagesInternational audienceSubmodular functions are set functions mapping every subset of some ground set of size into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time where denotes the cost to evaluate the function on any set. For functions with range , the best -additive approximation algorithm [CLSW17] runs in time . In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time . Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time . The main ingredient of the quantum result is a new method for sampling with high probability independent elements from any discrete probability distribution of support size in time . Previous quantum algorithms for this problem were of complexity
Quantum and Classical Algorithms for Approximate Submodular Function Minimization
22 pagesSubmodular functions are set functions mapping every subset of some ground set of size into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time where denotes the cost to evaluate the function on any set. For functions with range , the best -additive approximation algorithm [CLSW17] runs in time . In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time . Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time . The main ingredient of the quantum result is a new method for sampling with high probability independent elements from any discrete probability distribution of support size in time . Previous quantum algorithms for this problem were of complexity