Abstract.
The vertex separator (VS) problem in a graph G=(V,E) asks for a partition of V into nonempty subsets A, B, C such that there is no edge between A and B, and |C| is minimized subject to a bound on max{|A|,|B|}. We give a mixed integer programming formulation of the problem and investigate the vertex separator polytope (VSP), the convex hull of incidence vectors of vertex separators. Necessary and sufficient conditions are given for the VSP to be full dimensional. Central to our investigation is the relationship between separators and dominators. Several classes of valid inequalities are investigated, along with the conditions under which they are facet defining for the VSP. Some of our proofs combine in new ways projection with lifting.
In a companion paper we develop a branch-and-cut algorithm for the (VS) problem based on the inequalities discussed here, and report on computational experience with a wide variety of (VS) problems drawn from the literature and inspired by various applications.
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Balas, E., de Souza, C.C.: The Vertex Separation Problem: A Polyhedral Investigation. MSRR-670, GSIA, Carnegie Mellon University, September 2003; and IC-04-09, Institute of Computing, State University of Campinas, Brazil, 2004. Available on the web under, http://www.dcc.unicamp.br/ic-tr-ftp/2004/04-09.ps.gz.
Cormen, T.H., Leiserson, C.H., Rivest, R.L.: Introduction to Algorithms. MIT Press, 1990
Djidjev, H.N.: Partitioning Planar Graphs with Vertex Costs: Algorithms and Applications. Algorithmica 28, 51–75 (2000)
Garg, N., Saran, H., Vazirani, V.V.: Finding Separator Cuts in Planar Graphs Within Twice the Optimal. SIAM J. Computing 29, 159–179 (1999)
Hadley, G.: Linear Algebra. Addison-Wesley, 1965
Heath, M.T., Ng, E., Peyton, B.W.: Parallel Algorithms for Sparse Linear Systems. SIAM Review 33, 420–460 (1991)
Lipton, R.J., Tarjan, R.E.: A Separator Theorem for Planar Graphs. SIAM J. Appl. Math. 36, 177–189 (1979)
Lipton, R.J., Tarjan, R.E.: Applications of a Planar Separator Theorem. SIAM J. Comput. 9, 615–627 (1980)
Miller, G.L., Teng, S.H., Thorston, W., Vavasis, S.A.: Geometric Separators for Finite Element Meshes. SIAM J. Scientific Comput. 19, 364–386 (1998)
Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, 1988
de Souza, C.C., Balas, E.: The Vertex Separator Problem: Algorithms and Computations. Math. Programming 103, 609–637 (2005)
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Research supported by the National Science Foundation through grant #DMI-0098427 and by the Office of Naval Research through contract N00014-97-1-0196
Research supported by the Brazilian agencies FAPESP (grant 01/14205–6), CAPES (grant BEX 04444/02–2) and CNPq (grants 302588/02–7 and Pronex 664107/97–4)
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Balas, E., Souza, C. The vertex separator problem: a polyhedral investigation. Math. Program. 103, 583–608 (2005). https://doi.org/10.1007/s10107-005-0574-7
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DOI: https://doi.org/10.1007/s10107-005-0574-7