Multiscale approach for the network compression-friendly ordering

We present a fast multiscale approach for the network minimum logarithmic arrangement problem. This type of arrangement plays an important role in a network compression and fast node/link access operations. The algorithm is of linear complexity and exhibits good scalability which makes it practical and attractive for using on large-scale instances. Its effectiveness is demonstrated on a large set of real-life networks. These networks with corresponding best-known minimization results are suggested as an open benchmark for a research community to evaluate new methods for this problem

Multilevel Methods for Sparsification and Linear Arrangement Problems on Networks

The computation of network properties such as diameter, centrality indices, and paths on networks may become a major bottleneck in the analysis of network if the network is large. Scalable approximation algorithms, heuristics and structure preserving network sparsification methods play an important role in modern network analysis. In the first part of this thesis, we develop a robust network sparsification method that enables filtering of either, so called, long- and short-range edges or both. Edges are first ranked by their algebraic distances and then sampled. Furthermore, we also combine this method with a multilevel framework to provide a multilevel sparsification framework that can control the sparsification process at different coarse-grained resolutions. Experimental results demonstrate an effectiveness of the proposed methods without significant loss in a quality of computed network properties. In the second part of the thesis, we introduce asymmetric coarsening schemes for multilevel algorithms developed for linear arrangement problems. Effectiveness of the set of coarse variables, and the corresponding interpolation matrix is the central problem in any multigrid algorithm. We are pushing the boundaries of fast maximum weighted matching algorithms for coarsening schemes on graphs by introducing novel ideas for asymmetric coupling between coarse and fine variables of the problem

Multilevel Combinatorial Optimization Across Quantum Architectures

Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the Post Moore\u27s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this paper, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. In order to demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave\u27s quantum annealer and IBM\u27s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitudes larger than the current quantum hardware size and observe results comparable to state-of-the-art solvers