Spectral partitioning

Spectral partitioning methods are also termed as geometric representation methods, because at first the graphs are converted into a geometric representation, then well-known geometric heuristics are used to carry out the partitioning. Essentially, the success of a spectral partitioning algorithm depends on the quality and speed of the conversion and the geometric heuristic.

Hereby, I review the basic concepts of spectral partitioning by describing the orig-inal spectral bipartitioning technique and one of its complex descendants, which can handle multi-way partitioning of hypergraphs as well. Although, in the beginning, the spectral partitioning attracted a lot of research efforts, its performance was not proved to be superior to the alternative iterative techniques in case of large graphs [36]. Fur-thermore, the spectral technique is not flexible enough for real word problems as it is hard to integrate more complex objectives and constraints [37] into the geometric rep-resentation. As a consequence, the multilevel move-based techniques are considered better alternatives in modern circuit design technology.

The first spectral bipartitioning heuristics were motivated by the work of Fiedler [38]

and Hall [39] to partition ordinary graphs with the simple cutnet metric. The theory is based on the special properties of theLaplacian matrixof a graph.

The Laplacian matrix of aG(V, E)graph is a matrix with|V|x|V|size and defined as

L=D−A (4.7)

whereDis thedegree matrixof the graph containing the degrees of the vertices in its diagonal, andAis theadjacency matrixof the graph. Thedegreeof a vertex is defined as the number of the incident edges, while the value of ana_ij element of the adjacency matrix is the number of the edges connecting v_i andv_j vertices. The specialty of L is that the sum of its rows is always zero. The l_iidiagonal element corresponding to vertex v_i contains the maximum cutnet which can be related tov_i, while the other l_ij elements in the row indicates the benefit of puttingv_iandv_j to the same partition class.

The presented definitions can be naturally extended for weighted graphs as well.

The Laplacian quadratic form can be defined as

x^TLx (4.8)

where isxis a vector ofR^|V|. In the original bipartitioning scenario, only onexvector is considered, and it is imagined as a partitioning vector containing ones and zeros to describe one of the two clusters of the partitioning. In this case the quadratic form equals to the number of edges cut by the cluster (see Equation 4.9), as in the quadratic form, the number of uncut edges inside the cluster are subtracted from the maximum possible cutnet of the vertices residing in the cluster.

x^TLx=X

e∈E

(x(u)−x(v))² (4.9)

Observing Equation 4.9, it is obvious that it cannot be negative. Furthermore, for all constant vectors (which contain the same constant for each of their entries) the quadratic form equals to zero. In other words, the smallest eigenvalue ofLis always zero and constant vectors constitute the corresponding eigenspace. A constant vector can represent a trivial solution, when we put all the vertices into the same cluster.

For bipartitioning, the second smallest eigenvalue λ₂ (also called algebraic con-nectivity) and the corresponding eigenvector (also calledFiedler vector) is more inter-esting. It is easy to see that theλ₂ > 0if and only if the graph is connected and the x∈ R^|V|Fiedler vector gives a 1D representation of the vertices in which the squared distance of the endpoints of the edges are minimized (see Equation 4.9). Consequently, the representation can be used to split the graph e.g. at the largest gap between two ver-tices, however, there is no guarantee that the continuous solution closely approximates the solution of thex∈N^|V|discrete case.

4.3.2.2 Spectral partitioning with multiple eigenvectors

In the last decades of the 20th century, several improvements and extensions of the original spectral bipartitioning were proposed [28] to adapt the algorithm for multi-way hypergraph partitioning. One of the most successful modification was the work of Alpert [40] that proposed to use as many eigenvectors of the Laplacian as possible.

Hypergraphs were converted to ordinary graphs using theclique net model, in which each hyperedge is replaced by weighted edges connecting each pair of the vertices of the hyperedge, although it was proved earlier that there is no universal weighting scheme for the clique net model that can perfectly realize the cutnet metric [28].

The k-way partitions were represented by the indicator matrix, which is a |V|xk matrix containing the partitioning vectors mentioned before. Using the indicator ma-trix in the quadratic form, the cutnet metric appears in the trace of the result, and the following equation can be reached via theL = U∆U^T eigenvalue decomposition of L.

One of the main discoveries used in the method was that the partitioning prob-lem with theF_cutnetobjective can be mapped to amax-sum vector partitioning prob-lem. A k-way partitioning S^k = {S₁, S₂, ..., S_k} of n d-dimensional vectors Y = {y₁, y₂, ..., y_n}is defined askdisjoint subsets ofY, such thatS₁∪S₂∪...∪S_k =Y, while the max-sum objective is defined as

g(S^k) = X

Si∈S^k

kY_ik² where Y_i = X

yh∈S_i

y_h (4.11)

(Equation 4.10 and 4.11) are the same:

F_cutnet(G) =g(S^k) (4.12)

The proof can be constructed by observing the kY_ik² = P|V| j=1(p

λ_jU_j^TX_i)² equality, whereU_jis thej_thcolumn of matrixU, andX_iis thei_thcolumn of the indicator matrix X.

In [40] the standard k-means clustering method [41] was utilized to address the max-sum vector partitioning, however, as it is a maximization procedure, the task was reformulated to maximize the|V|λmax−Fcutnetobjective.