Theory and MethodologyAlgorithms for graph partitioning problems by means of eigenspace relaxations
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 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). Letbe 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 such thatwhere Û is the matrix in which attains the optimal Donath–Hoffman lower bound in Eq. (42). That is, andNote that Û can be computed by either semidefinite programming algorithms or subgradient algorithms. The problem of finding a with a given 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:
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Boppana's algorithm solves only equal-sized graph bisection problem and produces optimal bisections with high probability for certain connected, unweighted, undirected graphs;
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Boppana's bound is better than the Donath–Hoffman bound for
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- 1
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