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A new lower bound for the minimum linear arrangement of a graph. (English) Zbl 1341.05236
Liebling, Th. (ed.) et al., The IV Latin-American algorithms, graphs, and optimization symposium, Puerto Varas, Chile, November 25–29, 2007. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 30, 87-92 (2008).
Summary: Given a graph $$G=(V,E)$$ on $$n$$ vertices, the minimum linear arrangement problem (MinLA) calls for a one-to-one function $$\psi : V \to \{1,\ldots, n\}$$ which minimizes $$\sum_{\{i,j\}}E|\psi (i) - \psi (j)|$$. MinLA is strongly NP-hard and very difficult to solve to optimality in practice. One of the reasons for this difficulty is the lack of good lower bounds. In this paper, we take a polyhedral approach to MinLA. We associate an integer polyhedron with each graph $$G$$, and derive many classes of valid linear inequalities. It is shown that a cutting plane algorithm based on these inequalities can yield competitive lower bounds in a reasonable amount of time. A key to the success of our approach is that our linear programs contain only $$|E|$$ variables. We conclude showing computational results on benchmark graphs from literature.
For the entire collection see [Zbl 1137.05001].

MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 68Q25 Analysis of algorithms and problem complexity
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