Computing optimal embeddings for planar graphs

Abstract

We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. At IPCO’ 99 we presented a first characterization of the set of all possible embeddings of a given biconnected planar graph G by a system of linear inequalities. This system of linear inequalities can be constructed recursively using SPQR-trees and a new splitting operation. In general, this approach may not be practical in the presence of high degree vertices. In this paper, we present an improvement of the characterization which allows us to deal efficiently with high degree vertices using a separation procedure. The new characterization exposes the connection with the asymmetric traveling salesman problem thus giving an easy proof that it is NP-hard to optimize arbitrary objective functions over the set of combinatorial embeddings. Computational experiments on a set of over 11000 benchmark graphs show that we are able to solve the problem for graphs with 100 vertices in less than one second and that the necessary data structures for the optimization can be build in less than 12 seconds.