A New Lower Bound for the Minimum Linear Arrangement of a Graph

Given a graph G = ( V , E ) on n vertices, the Minimum Linear Arrangement Problem (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.