Abstract
The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000, which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n≤1000.
Highlights
Developed bounds for OPTGBP, which are based on semidefinite programming relaxations of the quadratic assignment problem or graph partitioning problem [40,44,45], involve semidefinite programs in matrices of order kn, which is much worse compared to our semidefinite programming problem (SDP)
It is well known that the ellipsoid method has very poor practical efficiency; we are interested in applying other, more efficient methods, such as interior-point methods [13,14,53] or the bundle method [15,16]
We report numerical results, obtained by Embed and Project Algorithm (EPA) on the test instances for which the problem GBPSDP is solvable by our implementation of the cutting-plane algorithm
Summary
A natural idea was to permute rows and columns of A such that all non-zero entries of the permuted matrix lie within a very narrow band along the main diagonal. This application gave the name to the problem: the matrix bandwidth problem (see, e.g., [3] and the references therein). The graph bandwidth problem (shortly, GBP) is the problem of finding the permutation of graph vertices such that the maximum difference of end point numbers, taken over all edges, is minimum: OPTGBP := min {max | φ(i ) − φ( j) | φ a permutation of V }.
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