A New Lower Bound for the Minimum Linear Arrangement of a Graph Andr´ e R.S. Amaral a Alberto Caprara b Adam N. Letchford c Juan-Jos´ e Salazar-Gonzalez d a Departamento de Inform´ atica, Universidade Federal do Esp´ ırito Santo, Vitoria, ES 29060-970, Brazil. E-mail: amaral@inf.ufes.br b DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. E-mail: acaprara@deis.unibo.it c Department of Management Science, Lancaster University, Lancaster LA1 4YX, Great Britain. E-mail: a.n.letchford@lancaster.ac.uk d DEIOC, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain. E-mail: jjsalaza@ull.es Abstract Given a graph G =(V,E) on n vertices, the Minimum Linear Arrangement Prob- lem (MinLA) calls for a one-to-one function ψ : V →{1,...,n} which minimizes ∑ {i,j }∈E |ψ(i) − ψ(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. Keywords: linear arrangement problem, polyhedral combinatorics, cutting planes Electronic Notes in Discrete Mathematics 30 (2008) 87–92 1571-0653/$ – see front matter © 2008 Published by Elsevier B.V. www.elsevier.com/locate/endm doi:10.1016/j.endm.2008.01.016