Abstract. We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.
Abstract. In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.1. Introduction. A general approach for solving many large-scale graph problems, as well as most other classes of large-scale computational science problems, is through multilevel (multiscale, multiresolution, etc.) algorithms. This approach generally involves coarsening the problem, producing from it a sequence of progressively coarser levels (smaller, hence simpler, related problems), then recursively using the (approximate) solution of each coarse problem to provide an initial approximation to the solution at the next-finer level. At each level, this initial approximation is first improved by what we generally call "local processing" (LP). This is an inexpensive sequence of short steps, each involving only a few unknowns, together covering all unknowns of that level several times over. The usual examples of LP are few sweeps of classical (e.g., Gauss-Seidel or Jacobi) relaxation in the case of solving a system of equations, or a few Monte Carlo passes in statistical-physics simulations. Following the LP, the resulting approximation may be further improved by one or several cycles, each using again a coarser-level approximation followed by LP, applying them at each time to the residual problem (the problem of calculating the error in the current approximation). See, for example, references [6,7,11,12,13,14,38,42].At each level of coarsening one needs to define the set of coarse unknown variables and the equations (or the stochastic relations) that they should satisfy (or the energy that they should minimize). Each coarse unknown is defined in terms of the nextfiner-level unknowns (defined, not calculated: they are all unknowns until the coarse level is approximately solved and the fine level is interpolated from that solution). The following are examples:• The set of coarse unknowns can simply represent a chosen subset of the finelevel set. • A coarse variable can be defined from several fine variables by a stochastic process ([5], for example).• In the case of graph problems, each node of the coarse graph can represent
Measuring the connection strength between a pair of vertices in a graph is one of the most vital concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the local connectivity. In this paper, we consider an iterative process that smoothes an associated value for nearby vertices, and present a measure of the connection strength (called the algebraic distance, see [28]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process confirms the underlying intuition that vertices are mutually reinforced and that the local neighborhoods play an important role in influencing the vertex connectivity. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
Hybrid quantum-classical algorithms such as the quantum approximate optimization algorithm (QAOA) are considered one of the most promising approaches for leveraging near-term quantum computers for practical applications. Such algorithms are often implemented in a variational form, combining classical optimization methods with a quantum machine to find parameters to maximize performance. The quality of the QAOA solution depends heavily on quality of the parameters produced by the classical optimizer. Moreover, the presence of multiple local optima in the space of parameters makes it harder for the classical optimizer. In this paper we study the use of a multistart optimization approach within a QAOA framework to improve the performance of quantum machines on important graph clustering problems. We also demonstrate that reusing the optimal parameters from similar problems can improve the performance of classical optimization methods, expanding on similar results for MAXCUT.
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