An efficient practical heuristic for good ratio-cut partitioning

Patkar, Sachin B.; Narayanan, Hariharan

doi:10.1109/icvd.2003.1183116

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…Because of the wide range of applications, several constraints and objective functions are considered. For instance, one can fix some vertices in clusters (like in the multicut problem), force equal-sized clusters etc., while optimizing (minimizing or maximizing) the sum of all edge weights between each pair of clusters (like in min-k-cut and max-k-cut), the sum of the edge weights (or the heaviest one) inside each cluster [8], or optimizing the cut ratio [12]. Some studies generalize many of these problems though one natural formalization: [7] gives computational lower bounds when the objective is to maximize some function over the inner edges of the clusters, [10] designs an O * (2 n ) algorithm for a whole class of partition problems such as max-k-cut, k-domatic partition or k-colouring, and [3] defines the M-partitioning problem where the objective is to find a partition of the vertices respecting some constraints defined by a matrix M .…”

Section: Related Workmentioning

confidence: 99%

“…Graph partitioning problems are the heart of many practical issues, especially for applications where some items must be grouped together, as in the design of VLSI layouts [11], clustering of social and biological networks [12], or software re-modularization [16]. Because of the wide range of applications, several constraints and objective functions are considered.…”

Section: Related Workmentioning

confidence: 99%

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On the sum-max graph partitioning problem

Watrigant

Bougeret

Giroudeau

et al. 2014

Theoretical Computer Science

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Abstract. In this paper we consider the classical combinatorial optimization graph partitioning problem, with Sum-Max as objective function. Given a weighted graph G = (V, E) and a integer k, our objective is to find a k-partition (V1, . . . , V k ) of V that minimizes, where w(u, v) denotes the weight of the edge {u, v} ∈ E. We establish the N P-completeness of the problem and its unweighted version, and the W [1]-hardness for the parameter k. Then, we study the problem for small values of k, and show the membership in P when k = 3, but the N P-hardness for all fixed k ≥ 4 if one vertex per cluster is fixed. Lastly, we present a natural greedy algorithm with an approximation ratio better than k 2 , and show that our analysis is tight.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

On the sum-max graph partitioning problem

Watrigant

Bougeret

Giroudeau

et al. 2014

Theoretical Computer Science

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“…Several heuristics were proposed, see e.g. [10,16,15,9] but they do not guarantee any upper bounds on the cut capacity. It is therefore common to consider a bicriteria approximation, which relaxes the balance constraint.…”

Section: Introductionmentioning

confidence: 99%

Multiply Balanced k −Partitioning

Amir

Ficler

Krauthgamer

et al. 2014

LATIN 2014: Theoretical Informatics

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Abstract. The problem of partitioning an edge-capacitated graph on n vertices into k balanced parts has been amply researched. Motivated by applications such as load balancing in distributed systems and market segmentation in social networks, we propose a new variant of the problem, called Multiply Balanced k Partitioning, where the vertex-partition must be balanced under d vertex-weight functions simultaneously. We design bicriteria approximation algorithms for this problem, i.e., they partition the vertices into up to k parts that are nearly balanced simultaneously for all weight functions, and their approximation factor for the capacity of cut edges matches the bounds known for a single weight function times d. For the case where d = 2, for vertex weights that are integers bounded by a polynomial in n and any fixed > 0, we obtain a (2 + , O( √ log n log k))-bicriteria approximation, namely, we partition the graph into parts whose weight is at most 2+ times that of a perfectly balanced part (simultaneously for both weight functions), and whose cut capacity is O( √ log n log k) · OPT. For unbounded (exponential) vertex weights, we achieve approximation (3, O(log n)). Our algorithm generalizes to d weight functions as follows: For vertex weights that are integers bounded by a polynomial in n and any fixed > 0, we obtain a (2d + , O(d √ log n log k))-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation (2d + 1, O(d log n)).

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“…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%

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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An efficient practical heuristic for good ratio-cut partitioning

Abstract: We present an efficient heuristic for finding good bipartitions of the vertex set of a graph in the sense of the wellknown measure of ratioCut [2,8]

Cited by 8 publications

References 13 publications

On the sum-max graph partitioning problem

On the sum-max graph partitioning problem

Multiply Balanced k −Partitioning

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

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