Fast Balanced Partitioning Is Hard Even on Grids and Trees

Feldmann, Andreas Emil

doi:10.1007/978-3-642-32589-2_34

Cited by 24 publications

(26 citation statements)

References 23 publications

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“…This problem is called Balanced Partitioning and is considerably harder than Bisection. For instance Balanced Partitioning is hard to approximate even on trees [15]. Nonetheless it is of great importance in applications such as parallel computing [3] and VLSI circuit design [4].…”

Section: Resultsmentioning

confidence: 99%

On the Parameterized Complexity of Computing Graph Bisections

Bevern

Feldmann

Sorge

et al. 2013

Graph-Theoretic Concepts in Computer Science

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Abstract. The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem.We show that Bisection is FPT w.r.t. the minimum cut size if there is an optimum bisection that cuts into a given constant number of connected components. Our algorithm applies to the more general Balanced Biseparator problem where vertices need to be removed instead of edges. We prove that this problem is W[1]-hard w.r.t. the minimum cut size and the number of cut out components.For Bisection we further show that no polynomial-size kernels exist for the cut size parameter. In fact, we show this for all parameters that are polynomial in the input size and that do not increase when taking disjoint unions of graphs. We prove fixed-parameter tractability for the distance to constant cliquewidth if we are given the deletion set. This implies fixed-parameter algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set.

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Section: Resultsmentioning

confidence: 99%

On the Parameterized Complexity of Computing Graph Bisections

Bevern

Feldmann

Sorge

et al. 2013

Graph-Theoretic Concepts in Computer Science

Self Cite

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“…In 5 (8) Values, of β close to one, indicate a well balanced partitioning. Bigger values of β indicate that the load is un-balanced.…”

Section: The Load Balancingmentioning

confidence: 98%

“…Therefore optimal processor load is w opt = W /p. The quality of partitioning can be then expressed [64] by the coefficient β β = max 0<i≤p w i w opt (8) Values, of β close to one, indicate a well balanced partitioning. Bigger values of β indicate that the load is un-balanced.…”

Section: The Load Balancingmentioning

confidence: 99%

“…This is achieved by equidistribution of the numerical load (represented by the distribution of mesh entities), and at the same time, by minimization of the number of cut edges. To this end, a range of partitioning methods, based on graph partitioning algorithms, have been developed [5][6][7][8]. Since adaptivity delivers a dynamic change to the distribution of mesh entities among processors a dynamic load balancing strategy must be used to maintain high performance of parallelization [9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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Parallel anisotropic mesh refinement with dynamic load balancing for transonic flow simulations

Gepner

Majewski

Rokicki

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The present paper discusses an effective adaptive methods suited for use in parallel environment. An in-house, parallel flow solver based on the residual distribution method is used for the solution of flow problems. Simulation is parallelized based on the domain decomposition approach. Adaptive changes to the mesh are achieved by two distinctive techniques. Mesh refinement is performed by dividing element edges and a subsequent application of pre defined splitting templates. Mesh regularization and derefinement is achieved through topology conserving node movement (r-adaptivity). Parallel implementations of an adaptive use the dynamic load balancing technique.Key words: parallel CFD, adaptation, dynamic load balancing, mesh refinement, parallel efficiency.Parallel anisotropic mesh refinement with dynamic load balancing for transonic flow simulations [28][29][30][31][32]. In general, this approach yields high quality meshes for arbitrarily complex geometries [4,[33][34][35][36]. Nevertheless this approach is difficult to implement in parallel simulations because of the global nature of the remeshing operation.An alternative approach is to apply modifications to the computational mesh only in the regions of interest. Either by local mesh modification operators (edge splits, edge collapses and edge swaps) [37][38][39][40] or element bisection (e.g., based on template refinement) [10,[41][42][43][44][45][46][47]. Locality of these methods makes them especially attractive for application in parallel computing, as the necessary communication volume and frequency become limited, boosting parallel performance.Another possibility is brought by the r-adaptive methods, in which grid nodes are allowed to move without changing the topology of the mesh. This technique is used either to augment local refinement methods [48,49], or as an adaptive tool in itself [50][51][52]. Application of r-adaptive algorithms might be beneficial in parallel computations, as neither the grid topology nor the load balance become affected.Although adaptive methods are well recognised in the academic community, the broader application of adaptation for industrial problems has been limited. This paper presents an approach towards fully parallel and automatic mesh adaptation method that can be used for 3D industrial cases. The h-r adaptive method is developed, suitable for parallel simulation of transonic flows. Adaptivity is accomplished by anisotropic template-based element bisection method. The coarsening step is carried out via an elastic analogy node movement. Anisotropic mesh modifications are driven by a Hessian based approach.

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“…Naively enumerating all partitions for a Markov chain Ca 2+ release site model and choosing the one with the smallest error is not possible because the number of valid partitions is too large. Actually, finding out how many ways one can divide a graph that possesses n vertices into k smaller components is know to be an NP-hard problem [22]. For example, a release site model composed of merely five three-state Ca 2+ channels (15 states) can be partitioned in approximately 10 10 distinct ways.…”

Section: Markov Chain Models Of Ca 2+ Signalingmentioning

confidence: 99%

Reduction of calcium release site models via optimized state aggregation

Hao

2016

EPJ Nonlinear Biomed Phys

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Background: Markov chain models of calcium release sites in living cells exhibit stochastic dynamics reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities for each of a large number of intercellular channel models depend on the local calcium concentration and thus the state of nearby channels. The state-space size in such compositionally defined calcium release site models increases exponentially as the number of channels increases, which is referred to as "state-space explosion". Methods: In order to overcome the state-space explosion problem, we utilized the idea of "coarse graining" and implemented an automated procedure that reduces the state space by aggregating and lumping states of the full release site model. For a given state aggregation scheme, the transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. A genetic algorithm-based approach is then applied to select the state aggregation schemes that lead to reduced models that approximate the observable behaviors of the full model. Results: The genetic algorithm-based approach is implemented in Matlab® and applied to two different release site models. The approach found reduced models that approximate the full model in the number of open channels, spark statistics, and the jump probability matrix as a function of time. Conclusions: A novel automated genetic algorithm-based searching technique is implemented to find reduced calcium release site models that approximate observable behaviors of the full Markov chain models that possess intractable state-spaces. As compared to the full model, the reduced models produce quantitatively similar results using significantly less computational resources.

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Fast Balanced Partitioning Is Hard Even on Grids and Trees

Cited by 24 publications

References 23 publications

On the Parameterized Complexity of Computing Graph Bisections

On the Parameterized Complexity of Computing Graph Bisections

Parallel anisotropic mesh refinement with dynamic load balancing for transonic flow simulations

Reduction of calcium release site models via optimized state aggregation

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