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AN EFFECTIVE REFINEMENT ALGORITHM BASED ON MULTILEVEL PARADIGM FOR GRAPH BIPARTITIONING

Ming Leng, Songnian Yu, Yang Chen School of Computer Engineering and Science,Shanghai University, Shanghai, PR China. Email: lmsly(3,263. net

Abstract:

The min-cut bipartitioning problem is a fundamental partitioning problem and is NP-Complete. It is also NP-Hard to find good approximate solutions for this problem. In this paper, we present a new effective refinement algorithm based on multilevel paradigm for graph bipartitioning. The success of our algorithm relies on exploiting both new Tabu search strategy and boundary refinement policy. Our experimental evaluations on 18 different graphs show that our algorithm produces excellent solutions compared with those produced by MeTiS that is a state-of-the-art partitioner in the literature.

Key words:

min-cut, graph bipartitioning, multilevel paradigm. Tabu search

1.

INTRODUCTION

Partitioning is a fundamental problem with extensive applications to many areas, including VLSI design [1], information retrieval [2], parallel processing [3], computational grids [4] and data mining [5]. The min-cut bipartitioning problem is a fundamental partitioning problem and is NPComplete [6]. It is also NP-Hard [7] to find good approximate solutions for this problem. Because of its importance, the problem has attracted a considerable amount of research interest and a variety of algorithms have been developed over the last thirty years [8],[9]. The survey by Alpert and Kahng [1] provides a detailed description and comparison of various such

Please use the foil owing format when citing this chapter: Leng, Ming, Yu, Songnian, Chen, Yang, 2006, in International Federation for Information Processing (IFIP), Volume 207, Knowledge Enterprise: Intelligent Strategies In Product Design, Manufacturing, and Management, eds. K. Wang, Kovacs G., Wozny M., Fang M., (Boston: Springer), pp. 294-303.

An Effective Refinement Algorithm Based on Multilevel Paradigm for Graph Bipartitioning

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schemes which include move-based approaches, geometric representations, combinatorial formulations, and clustering approaches. As problem sizes reach new levels of complexity recently, a new class of graph partitioning algorithms have been developed that are based on the multilevel paradigm. The multilevel graph partitioning schemes include three phases [10], [11],[12], [13]. The coarsening phase is to reduce the size of the graph by collapsing vertex and edge until its size is smaller than a given threshold. The initial partitioning phase is to compute initial partition of the coarsest graph. The uncoarsening phase is to successively project the partition of the smaller graph back to the next level finer graph while applying an iterative refinement algorithm. In this paper, we present a new effective refinement algorithm based on multilevel paradigm which is integrated with Tabu search [14] for refining the partition. Our work is motivated by the multilevel partitioners of Saab [13] who promotes locked vertex to free by introducing two buckets for per side of the partition and Karypis [10], [11],[12] who proposes a boundary refinement algorithm and supplies MeTiS [10], distributed as open source software package for partitioning unstructured graphs. We test our algorithm on 18 graphs that are converted from the hypergraphs of the ISPD98 benchmark suite [15]. Our experiments show that our algorithm produces partitions that are better than those produced by MeTiS in a reasonable time. The rest of the paper is organized as follows. Section 2 provides some definitions, describes the notation that is used throughout the paper and the min-cut bipartitioning problem. Section 3 briefly describes the motivation behind our algorithm. Section 4 presents an effective refinement algorithm based on multilevel paradigm for graph bipartitioning. Section 5 experimentally evaluates the algorithm and compares it with MeTiS. Finally, Section 6 provides some concluding remarks and indicates the directions for fiirther research.

2.

MATHEMATICAL DESCRIPTION

A graph G = (V,E) consists of a set of vertices V and a set of edges E such that each edge is a subset of two vertices in V. Throughout this paper, n and m denote the number of vertices and edges respectively. The vertices are numbered from I ton and each vertex v e v has an integer weight S{v) . The edges are numbered from I to m and each edge eeE has an integer weight w{e) .A decomposition of a graph V into two disjoint subsets Vi and V2, such that v^[jV2 = v and v;n\/2 = 0 , is called a bipartitioning of V. Let

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Ming Leng, Songnian Yu, Yang Chen

s(A) = 2^^^s(v) denote the size of a subset A^v . Let /D(v) be denoted as v's internal degree and is equal to the sum of the edge-weights of the adjacent vertices of v that are in the same side of the partition as v, and v's external degree denoted by ED{\/) is equal to the sum of edge-weights of the adjacent vertices of v that are in the different side of the partition. The cut of a bipartitioning P = ^^y2} is the sum of weights of edges which contain two vertices in Vi and V2 respectively, such that cut(P) = ^^^^^^^^^^^^w{e) . Naturally, vertex v belongs at boundary if and only if ED{v) > 0 and the cut of P is also equal to 0.5^,^^^ ED(V) . Given a balance constraint r, the min-cut bipartitioning problem seeks a solution P = ^v^2} that minimizes cut{P) subject to (i-r)S(\/)/2