Theory and Methodology
Algorithms for graph partitioning problems by means of eigenspace relaxations

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Abstract

Graph partitioning problems are NP-hard problems and very important in VLSI design. We study relations among several eigenvalue bounds and algorithms for graph partitioning problems. Also, we design an algorithm for the problems which performs the following: first it computes the k largest eigenvalues of the affine symmetric matrix function to attain Donath–Hoffman bound; then it calculates a relaxed partition which is an array constant factor of an eigenspace associated with k eigenvalues; finally it generates an actual partition from the relaxed solution of a method similar to Boppana's algorithm. To compute optimal eigenvalue bounds, one needs to solve eigenvalue optimization problems which minimize the sum of the k largest eigenvalues of the nonsmooth functions. We use a subgradient method to compute the Donath–Hoffman eigenvalue bound. Numerical results indicate that although the Donath–Hoffman bound is not tight for graph partitioning problems, our algorithm can generate optimal partitions.

Introduction

Graph partitioning problems are concerned with partitioning the set of vertices of a given graph into k disjoint subsets. Each subset has a specified size so as to minimize the total weight of edges that connect distinct subsets. These problems are NP-hard problems (see [11]) and important in VLSI design (see [17]). Several heuristic algorithms perform well in practice (see 16, 1, 3, 13, 6, 2, 19, 20) and have been shown to have good average case behavior over certain probability distributions on graphs (see 5, 4). We are interested in the application of spectral methods which are related to eigenvalue bounds and eigenvector space utilizations (see 9, 1, 3, 4, 5, 20, 10).

In Section 2, we investigate the relations among the Donath–Hoffman eigenvalue bound [9], Barnes' algorithm 3, 6, Boppana's eigenvalue bound [4] and Rendl–Wolkowicz's projection model [20]. In Section 3, we review the Donath–Hoffman eigenvalue bound for graph partitioning problems. In Section 4, we design a partitioning algorithm which extends the combination of Barnes' and Boppana's algorithm. For a graph partitioning problem, first, our algorithm computes k eigenvalues to attain the optimal Donath–Hoffman bound; then, it generates a relaxed partition which is the eigenspace associated with k eigenvalues; finally, an actual partition is computed by using a method, similar to Boppana' algorithm, on the relaxed partition, i.e. the eigenspace. Again, to compute the optimal Donath–Hoffman eigenvalue bound, one needs to solve an eigenvalue optimization problem which minimizes the sum of the k largest eigenvalues of the affine symmetric matrix function. We solve this problem with a subgradient method from Cullum and Donath [7]. In Section 4.1, we test our algorithm with some graphs for bisection problems, i.e. equal-sized bisection and nonequal-sized bisection problems. In Section 4.2, we solve an equal-sized graph trisection problem.

Section snippets

Preliminary

Consider a connected, weighted, and undirected graph (V,E,G) without any self-loop of nodes, where V is the set of nodes, E is the set of edges of the graph, and G is the associated adjacency matrix. Therefore, Gij=0 if i=j or {i,j} is not an edge in E and Gij represents the positive weight of the edge {i,j} if {i,j}∈E. Define |V|=n to be the number of nodes, |E|=sum(G)/2 to be the sum of the weights of all edges, where sum(G) is equal to the sum of all n2 entries of the adjacency matrix G. Let

The Donath–Hoffman eigenvalue bound for the graph partitioning problem

We are given m=(m1,m2,…,mk)TRk the specified size of a partition and G the adjacency matrix of the weighted graph (V,E,G). Let sum(G) be the sum of all n2 entries of the adjacency matrix G and sum(G)/2 be the sum of the weights of all edges of the graph (V,E,G). Letsum(G)2maxX∈Xnm12tr(XTGX),be the sum of the weights of edges that connect distinct subsets, Vi, i=1,…,k, which are represented by the optimal partition. Let g(m,G,U) be the function of the sum of the weights of edges that connect

Algorithms for the graph partitioning problem

Our graph partitioning algorithm is to find a P̂Pnm such that||(G+Û)−P̂||2=minP∈Pnm||(G+Û)−P||2,where Û is the matrix in K which attains the optimal Donath–Hoffman lower bound in Eq. (42). That is, ÛK andη(m,G,U)=sum(G)212minU∈Ki=1kmiλi(G+U),=sum(G)212i=1kmiλi(G+Û).Note that Û can be computed by either semidefinite programming algorithms or subgradient algorithms. The problem of finding a P̂Pnm with a given ÛK has been studied by Barnes et al. [6]. Here, we propose another approach

Conclusions

By applying Lemma 1 and Eq. (18), one can solve a quadratic assignment problem by mapping the problem itself into a graph partitioning problem. Algorithm 3 solves a general graph partitioning problem. Numerical results in previous sections indicate the following:

  • Boppana's algorithm solves only equal-sized graph bisection problem and produces optimal bisections with high probability for certain connected, unweighted, undirected graphs;

  • Boppana's bound is better than the Donath–Hoffman bound for

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