Hypergraph partitioning is an NP-hard problem that occurs in many computer science applications where it is necessary to reduce large problems into a number of smaller, computationally tractable sub-problems. Current techniques use a multilevel approach wherein an initial partitioning is performed after compressing the hypergraph to a predetermined level. This level is typically chosen to produce very coarse hypergraphs in which heuristic algorithms are fast and effective. This article presents a novel memetic algorithm which remains effective on larger initial hypergraphs. This enables the exploitation of information that can be lost during coarsening and results in improved final solution quality. We use this algorithm to present an empirical analysis of the space of possible initial hypergraphs in terms of its searchability at different levels of coarsening. We find that the best results arise at coarsening levels unique to each hypergraph. Based on this, we introduce an adaptive scheme that stops coarsening when the rate of information loss in a hypergraph becomes non-linear and show that this produces further improvements. The results show that we have identified a valuable role for evolutionary algorithms within the current state-of-the-art hypergraph partitioning framework.
Many EAs have been applied to the more well-known problem of graph partitioning; see Kim et al. @cite_32 for an overview. Soper et al. @cite_9 were the first to use an EA within a multilevel approach. They introduced variation operators that modify the edge weights of the graph depending on the input partitions. Subsequently presenting these to a multilevel partitioner, which uses the weights to obtain a new partition.
@cite_9 @cite_32
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Graph partitioning is one of the most studied NP-complete problems. Given a graph G=(V, E) , the task is to partition the vertex set V into k disjoint subsets of about the same size, such that the number of edges with endpoints in different subsets is minimized. In this paper, we present a highly effective multilevel memetic algorithm, which integrates a new multiparent crossover operator and a powerful perturbation-based tabu search algorithm. The proposed crossover operator tends to preserve the backbone with respect to a certain number of parent individuals, i.e., the grouping of vertices which is common to all parent individuals. Extensive experimental studies on numerous benchmark instances from the graph partitioning archive show that the proposed approach, within a time limit ranging from several minutes to several hours, performs far better than any of the existing graph partitioning algorithms in terms of solution quality. This paper presents a new spectral partitioning formulation which directly incorporates vertex size information by modifying the Laplacian of the graph. Modifying the Laplacian produces a generalized eigenvalue problem, which is reduced to the standard eigenvalue problem. Experiments show that the scaled ratio-cut costs of results on benchmarks with arbitrary vertex size improve by 22 when the eigenvectors of the Laplacian in the spectral partitioner KP are replaced by the eigenvectors of our modified Laplacian. The inability to handle vertex sizes in the spectral partitioning formulation has been a limitation in applying spectral partitioning in a multilevel setting. We investigate whether our new formulation effectively removes this limitation by combining it with a simple multilevel bottom-up clustering algorithm and an iterative improvement algorithm for partition refinement. Experiments show that in a multilevel setting where the spectral partitioner KP provides the initial partitions of the most contracted graph, using the modified Laplacian in place of the standard Laplacian is more efficient and more effective in the partitioning of graphs with arbitrary-size and unit-size vertices; average improvements of 17 and 18 are observed for graphs with arbitrary-size and unit-size vertices, respectively. Comparisons with other ratio-cut based partitioners on hypergraphs with unit-size as well as arbitrary-size vertices, show that the multilevel spectral partitioner produces either better results or almost identical results more efficiently. The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages. The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy is in contrast to that of the widely used spectral bipartitioning (SB) heuristic (which uses only a single eigenvector) and several previous multi-way partitioning heuristics [8, 11, 17, 27, 38] (which use k eigenvectors to construct k-way partitionings). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only significantly outperforms recursive SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1, 11]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective partitioning heuristics. The present paper updates and improves a preliminary version of this work [5].