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.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bian, Z., Gu, Q.P.: Computing Branch Decomposition of Large Planar Graphs. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 87–100. Springer, Heidelberg (2008)
Bian, Z., Gu, Q.P., Marzban, M., Tamaki, H., Yoshitake, Y.: Empirical study on branchwidth and branch decomposition of planar graphs. In: Proc. of the 9th SIAM Workshop on Algorithm Engineering and Experiments (ALENEX 2008), pp. 152–165 (2008)
Bodlaender, H.L., Grigoriev, A., Koster, A.M.C.A.: Treewidth lower bounds with brambles. Algorithmica 51(1), 81–89 (2008)
Byers, T.H., Waterman, M.S.: Determining all optimal and near-optimal solutions when solving shortest path problems by dynamic programming. Operations Research 32(6), 1381–1384 (1984)
de Fraysseix, H., de Mendez, P.O.: PIGALE-Public Implementation of a Graph Algorithm Library and Editor. SourceForge project page, http://sourceforge.net/projects/pigale
Demaine, E., Fomin, F., Hajiaghayi, M., Thilikos, D.: Bidimensional parameters and local treewidth. SIAM J. Discret. Math. 18(3), 501–511 (2005)
Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. Journal of the ACM (JACM) 52(6), 866–893 (2005)
Demaine, E.D., Hajiaghayi, M.T.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28(1), 19–36 (2008)
Demaine, E.D., Hajiaghayi, M.T., Thilikos, D.M.: The bidimensional theory of bounded-genus graphs. SIAM Journal on Discrete Mathematics 20(2), 357–371 (2007)
Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. Algorithmica 58(3), 790–810 (2010)
Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976)
Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Gu, Q.P., Tamaki, H.: Improved bounds on the planar branchwidth with respect to the largest grid minor size. Algorithmica (to appear, 2011)
Gu, Q.P., Tamaki, H.: Optimal branch-decomposition of planar graphs in O(n 3) time. ACM Transactions on Algorithms (TALG) 4(3), 1–13 (2008)
Marzban, M., Gu, Q.P., Jia, X.: Computational study on planar dominating set problem. Theoretical Computer Science 410(52), 5455–5466 (2009)
Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press (1999)
Reinelt, G.: TSPLIB–A traveling salesman problem library. INFORMS Journal on Computing 3(4), 376 (1991)
Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a planar graph. Journal of Combinatorial Theory, Series B 62(2), 323–348 (1994)
Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B 52(2), 153–190 (1991)
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)
Wang, C.: Computational study on bidimensionality theory based algorithms. MSc Thesis, Simon Fraser University (August 2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wang, C., Gu, QP. (2011). Computational Study on Bidimensionality Theory Based Algorithm for Longest Path Problem. In: Asano, T., Nakano, Si., Okamoto, Y., Watanabe, O. (eds) Algorithms and Computation. ISAAC 2011. Lecture Notes in Computer Science, vol 7074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25591-5_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-25591-5_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25590-8
Online ISBN: 978-3-642-25591-5
eBook Packages: Computer ScienceComputer Science (R0)