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A multilevel bilinear programming algorithm for the vertex separator problem. (English) Zbl 1394.90546
Summary: The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper [the first two authors, Eur. J. Oper. Res. 240, No. 2, 328–337 (2015; Zbl 1357.90166)] as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. We develop a method for improving upon a given vertex separator by applying a Mountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm can improve upon the starting guess after at most two iterations. The refinement algorithm is augmented with a perturbation technique to enable escapes from local optima and is embedded in a multilevel framework for solving large scale instances of the problem. The multilevel algorithm is shown through computational experiments to perform particularly well on communication and collaboration networks.

90C35 Programming involving graphs or networks
90C27 Combinatorial optimization
90C20 Quadratic programming
90C06 Large-scale problems in mathematical programming
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