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 ∑{|ψ(i)−ψ(j)| : {i,j} in E}. MinLA is strongly -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.
A.R.S. Amaral, A. Caprara, A.N. Letchford, J.J. Salazar (2008). A New Lower Bound for the Minimum Linear Arrangement of a Graph. ELECTRONIC NOTES IN DISCRETE MATHEMATICS, 30, 87-92 [10.1016/j.endm.2008.01.016].
A New Lower Bound for the Minimum Linear Arrangement of a Graph
CAPRARA, ALBERTO;
2008
Abstract
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 ∑{|ψ(i)−ψ(j)| : {i,j} in E}. MinLA is strongly -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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
