Abstract
This paper presents the analysis of the 2-sum problem and the spectral algorithm. The spectral algorithm was proposed by Barnard, Pothen and Simon in [1]; its heuristic properties have been advocated by George and Pothen in [4] by formulation of the 2-sum problem as a Quadratic Assignment Problem. In contrast to that analysis another approach is proposed: permutations are considered as vectors of Euclidian space. This approach enables one to prove the bound results originally obtained in [4] in an easier way. The geometry of permutations is considered in order to explain what are ‘good’ and ‘pathological’ situations for the spectral algorithm. Upper bounds for approximate solutions generated by the spectral algorithm are proved. The results of numerical computations on (graphs of) large sparse matrices from real-world applications are presented to support the obtained results and illustrate considerations related to the ‘pathological’ cases.
Highlights
The 2-sum problem is one of the graph enumeration problems
Pothen and Simon proposed spectral algorithm for envelope reduction of sparse matrices [1]. Their algorithm yields an approximate solution of a 2-sum problem, which itself is used as an approximation for reducing envelope and envelope-related parameters of large sparse matrices
Spectral algorithm provides a good balance between the computational cost and the accuracy of the approximate solution obtained for NP-complete 2-sum problem
Summary
The 2-sum problem (to be defined in section 2) is one of the graph enumeration problems ( known as graph layout problems). Fiedler studied the properties of the second Laplacian eigenvalue and eigenvector ( called Fiedler vector, i.e. eigenvector of Laplacian of a connected graph that corresponds to the second smallest eigenvalue) He observed that the differences between the components of this eigenvector are an approximate measure of the distance between the vertices [2,3]. An approximate solution computed by spectral algorithm, generally, is only a heuristic approximation as the 2-sum problem was proved to be NP-complete [4]. The idea is to consider the related 2-sum problem, and show that a second Laplacian eigenvector x2 solves a continuous relaxation of the problem. For odd n , let T denote the set of vectors whose components are permutations of. The heuristic idea of choosing the permutation closest to the accurate solution naturally has to be such that, given the close position of the permutation chosen to the accurate solution, the increase of the cost function (quadratic form of matrix Q ) remains small enough
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