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A practical optimization framework for political redistricting: A case study in Arizona

SOCIO-ECONOMIC PLANNING SCIENCES 2024年 92卷

作者： Swamy, Rahul King, Douglas M. Ludden, Ian G. Dobbs, Kiera W. Jacobson, Sheldon H. Univ Illinois Dept Ind & Enterprise Syst Engn 104 S Mathews Ave Urbana IL 61801 USA Rose Hulman Inst Technol Dept Comp Sci & Software Engn 5500 Wabash Ave Terre Haute IN 47803 USA Univ Illinois Dept Comp Sci 201 N Goodwin Ave Urbana IL 61801 USA

Following the decennial census, each state in the U.S. redraws its congressional and state legislative district boundaries, which must satisfy various legal criteria. For example, Arizona's Constitution describes six legal criteria, including contiguity, population balance, competitiveness, compactness, and the preservation of communities of interest, political subdivisions and majority-minority districts, each of which is to be enforced "to the extent practicable". Optimization algorithms are well suited to draw district maps, although existing models and methods have limitations that inhibit their ability to draw legally -valid maps. Adapting existing optimization methods presents two major challenges: the complexity of modeling to achieve multiple and subjective criteria, and the computational intractability when dealing with large redistricting input graphs. In this paper, we present a multi -stage optimization framework tailored to redistricting in Arizona. This framework combines key features from existing methods, such as a multilevel algorithm that reduces graph input sizes and a larger local search neighborhood that encourages faster exploration of the solution space. This framework heuristically optimizes geographical compactness and political competitiveness while ensuring that other criteria in Arizona's Constitution are satisfied relative to existing norms. Compared to Arizona's enacted map (CD118) to be used until 2032, the most compact map produced by the algorithm is 41% more compact, and the most competitive map has five more competitive districts. To enable accessibility and to promote future research, we have created Optimap, a publicly accessible tool to interact with a part of this framework. Beyond the creation of these maps, this case study demonstrates the positive impact of adapting optimization -based methodologies for political redistricting in practice.

关键词： Political redistricting Optimization algorithms Local search multilevel algorithms Arizona Gerrymandering

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Numerical Solution of Markov Chains

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2011年第6期18卷 897-900页

作者： Benzi, Michele Emory Univ Dept Math & Comp Sci Atlanta GA 30322 USA

This special issue contains a selection of papers from the Sixth International Workshop on the Numerical Solution of Markov Chains, held in Williamsburg, Virginia on September 16-17, 2010. The papers cover a broad range of topics including perturbation theory for absorbing chains, bounding techniques, steady-state and transient solution methods, multilevel algorithms, preconditioning, and applications. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： Markov chains multilevel algorithms preconditioning applied probability

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multilevel graph embedding

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2021年第3期28卷 e2326-e2326页

作者： Quiring, Benjamin Vassilevski, Panayot S. Northeastern Univ Dept Comp Sci Boston MA 02115 USA Lawrence Livermore Natl Lab Ctr Appl Sci Comp POB 808L-561 Livermore CA 94551 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA

The goal of the present paper is the design of embeddings of a general sparse graph into a set of points inDouble-struck capital Rdfor appropriated >= 2. The embeddings that we are looking at aim to keep vertices that are grouped in communities together and keep the rest apart. To achieve this property, we utilize coarsening that respects possible community structures of the given graph. We employ a hierarchical multilevel coarsening approach that identifies communities (strongly connected groups of vertices) at every level. The multilevel strategy allows any given (presumably expensive) graph embedding algorithm to be made into a more scalable (and faster) algorithm. We demonstrate the presented approach on a number of given embedding algorithms and large-scale graphs and achieve speed-up over the methods in a recent paper.

关键词： embeddings graphs multilevel algorithms

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Low-Rank Approximation of Impedance Matrix Blocks Using Non-uniform Sampling Methods

Low-Rank Approximation of Impedance Matrix Blocks Using Non-...

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IEEE International Symposium on Antennas and Propagation / USNC-URSI Radio Science Meeting

作者： Brick, Yaniv Yilmaz, Ali E. Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

ISBN: (纸本)9781509028528

Two physics-based rank-revealing multilevel algorithms are presented to efficiently compute impedance matrix blocks' low-rank representations. Both algorithms rely on non-uniform sampling of phase-and amplitude-compensated fields but use different auxiliary grids. The algorithms' computational costs and range of validity are compared.

关键词： direct solves fast methods multilevel algorithms

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Infinite-Variate L²-Approximation with Nested Subspace Sampling 15th

Infinite-Variate L<SUP>2</SUP>-Approximation with Nested Sub...

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15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC)

作者： Harsha, Kumar Gnewuch, Michael Wnuk, Marcin Osnabruck Univ Inst Math Albrechtrasse 28a D-49074 Osnabruck Germany

ISBN: (纸本)9783031597640;9783031597626;9783031597619

We consider L-2-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear functionals. We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces, we refer to function spaces whose norms are induced by an underlying ANOVA function decomposition. In ANOVA spaces, we provide an optimal algorithm to solve the approximation problem using linear information. We determine the upper and lower error bounds on the polynomial convergence rate of n-th minimal worst-case errors, which match if the weights decay regularly. For non-ANOVA spaces, we also establish upper and lower error bounds. Our analysis reveals that for weights with a regular and moderate decay behavior, the convergence rate of n-th minimal errors is strictly higher in ANOVA than in non-ANOVA spaces.

关键词： ANOVA decomposition Information-based complexity Reproducing kernel hilbert space weighted function spaces multilevel algorithms

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A multilevel approach for overlapping community detection 3

A multilevel approach for overlapping community detection

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3rd Brazilian Conference on Intelligent Systems (BRACIS)

作者： Valejo, Alan Valverde-Rebaza, Jorge Lopes, Alneu de Andrade Univ Sao Paulo ICMC Dept Comp Sci BR-13560970 Sao Carlos SP Brazil

ISBN: (纸本)9781479978595

Many real world complex networks have an a overlapping community structure, in which a vertex belongs to one or more communities. Numerous approaches for crisp overlapping community detection were proposed in the literature, most of them have a good accuracy but their computational costs are considerably high and infeasible for large-scale networks. Since the multilevel approach has not been previously applied to deal with overlapping communities detection problem, in this paper we propose an adaptation of this approach to tackle the detection problem to overlapping communities case. The goal is to analyze the time impact and the quality of solution of our multilevel strategy regarding to traditional algorithms. Our experiments show that our proposal consistently produces good performance compared to single-level algorithms and in less time.

关键词： Complex networks Community detection Overlapping community multilevel algorithms

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A cooperative multilevel tabu search algorithm for the covering design problem

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7th International Conference on Evolution Artificielle

作者： Dai, Chaoying Li, (Ben) Pak Ching Toulouse, Michel Univ Manitoba Dept Comp Sci Winnipeg MB R3T 2N2 Canada

ISBN: (纸本)3540335897

This work describes an adaptation of multilevel search to the covering design problem. The search engine is a tabu search algorithm which explores several levels of overlapping search spaces of a t-(v, k, lambda) covering design problem. Tabu search finds "good" approximations of covering designs in each search space. Blocks from those approximate solutions are transferred to other levels, redefining the corresponding search spaces. The dynamics of cooperation among levels tends to regroup good approximate solutions into small search spaces. Tabu search has been quite effective at finding re-combinations of blocks in small search spaces which provide successful search directions in larger search spaces.

关键词： multilevel algorithms covering design problem tabu search meta-heuristic

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Accelerated Randomized vs. Deterministic algorithms for MoM Blocks' Low-Rank Factorization

Accelerated Randomized vs. Deterministic Algorithms for MoM ...

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International Symposium of IEEE-Antennas-and-Propagation-Society / USNC/URSI National Radio Science Meeting

作者： Brick, Yaniv Yilmaz, Ali E. Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

ISBN: (纸本)9781538632840

A randomized and a deterministic algorithm for the fast computation of impedance matrix blocks' low-rank factorization are contrasted. The deterministic algorithm is based on multilevel compression of non-uniformly sampled phase-and amplitude-compensated interactions between clusters of source/observer basis/testing functions. The randomized algorithm is accelerated by using fast Fourier transform-based field evaluation. The algorithms' performance are compared for various classes of problem topology and algorithm parameters.

关键词： direct solves fast methods multilevel algorithms

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Gradient-Based Simulation Optimization algorithms via Multi-Resolution System Approximations

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INFORMS JOURNAL ON COMPUTING 2023年第3期35卷 633-651页

作者： Xu, Jingxu Zheng, Zeyu Univ Calif Berkeley Dept Ind Engn & Operat Res Berkeley CA 94720 USA

We propose gradient-based simulation-optimization algorithms to optimize systems that have complicated stochastic structure. The presence of complicated stochastic structure, such as the involvement of infinite-dimensional continuous-time stochastic processes, may cause the exact simulation of the system to be costly or even impossible. On the other hand, for a complicated system, one can sometimes construct a sequence of approximations at different resolutions, where the sequence has finer and finer approximation resolution but higher and higher cost to simulate. With the goal of optimizing the complicated system, we propose algorithms that strategically use the approximations with increasing resolution and higher simulation cost to construct stochastic gradients and perform gradient search in the decision space. To accommodate scenarios where approximations cause discontinuities and lead path-wise gradient estimators to have an uncontrollable bias, stochastic gradients for the proposed algorithms are constructed through finite difference. As a theory support, we prove algorithm convergence, convergence rate, and optimality of algorithm design under the assumption that the objective function for the complicated system is strongly convex, whereas no such assumptions are imposed on the approximations of the complicated system. We then present a multilevel version of the proposed algorithms to further improve convergence rates, when in addition the sequence of approximations can be naturally coupled.

关键词： gradient-based simulation-optimization algorithms approximating systems multilevel algorithms convergence rates

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A Class of Symmetric Factored Approximate Inverses and Hybrid Two-Level Solver

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2018年第6期15卷

作者： Kyziropoulos, Panagiotis E. Filelis-Papadopoulos, Christos K. Gravvanis, George A. Democritus Univ Thrace Sch Engn Dept Elect & Comp Engn Univ Campus GR-67100 Xanthi Greece

A new class of symmetric factored approximate inverses is proposed and used in conjunction with the Preconditioned Conjugate Gradient method for solving sparse symmetric linear systems. Additionally, a new hybrid two-level solver is proposed utilizing a block independent set reordering, in order to create the two level hierarchy. The Schur complement is formed explicitly by inverting the blocks created by reordering. Then, the preconditioned conjugate gradient method is used in conjunction with the symmetric factored approximate inverse to solve the reduced order linear system. Furthermore, numerical results on the performance and convergence behavior for solving various model problems are presented.

关键词： Symmetric factored approximate sparse inverses two level algebraic symmetric factored approximate inverse solver multilevel algorithms preconditioned conjugate gradient

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