Computational Study on Bidimensionality Theory Based Algorithm for Longest Path Problem

Abstract

Bidimensionality theory provides a general framework for developing subexponential fixed parameter algorithms for NP-hard problems. In this framework, to solve an optimization problem in a graph G, the branchwidth ${\mathop{\rm bw}}(G)$ is first computed or estimated. If ${\mathop{\rm bw}}(G)$ is small then the problem is solved by a branch-decomposition based algorithm which typically runs in polynomial time in the size of G but in exponential time in ${\mathop{\rm bw}}(G)$ . Otherwise, a large ${\mathop{\rm bw}}(G)$ implies a large grid minor of G and the problem is computed or estimated based on the grid minor. A representative example of such algorithms is the one for the longest path problem in planar graphs. Although many subexponential fixed parameter algorithms have been developed based on bidimensionality theory, little is known on the practical performance of these algorithms. We report a computational study on the practical performance of a bidimensionality theory based algorithm for the longest path problem in planar graphs. The results show that the algorithm is practical for computing/estimating the longest path in a planar graph. The tools developed and data obtained in this study may be useful in other bidimensional algorithm studies.