Abstract
The problem of determining the cutwidth of a graph is a notoriously hard problem which remains NP-complete under severe restrictions on input graphs. Until recently, non-trivial polynomial-time cutwidth algorithms were known only for subclasses of graphs of bounded treewidth. In WG 2008, Heggernes et al. initiated the study of cutwidth on graph classes containing graphs of unbounded treewidth, and showed that a greedy algorithm computes the cutwidth of threshold graphs. We continue this line of research and present the first polynomial-time algorithm for computing the cutwidth of bipartite permutation graphs. Our algorithm runs in linear time. We stress that the cutwidth problem is NP-complete on bipartite graphs and its computational complexity is open even on small subclasses of permutation graphs, such as trivially perfect graphs.
This work is supported by the Research Council of Norway and by EPSRC UK grant EP/D053633/1.
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References
Adolphson, D., Hu, T.C.: Optimal linear ordering. SIAM J. Appl. Math. 25, 403–423 (1973)
Arora, S., Frieze, A., Kaplan, H.: A new rounding procedure for the assignment problem with applications to dense graphs arrangements. In: Proceedings of FOCS 1996, pp. 21–30. IEEE, Los Alamitos (1996)
Blin, G., Fertin, G., Hermelin, D., Vialette, S.: Fixed-parameter algorithms for protein similarity search under RNA structure constraints. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 271–282. Springer, Heidelberg (2005)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Brandstädt, A., Lozin, V.V.: On the linear structure and clique-width of bipartite permutation graphs. Ars Combinatorica 67, 273–289 (2003)
Chung, M.J., Makedon, F., Sudborough, I.H., Turner, J.: Polynomial time algorithms for the min cut problem on degree restricted d trees. In: Proceedings of FOCS 1982, pp. 262–271. IEEE, Los Alamitos (1982)
Díaz, J., Penrose, M., Petit, J., Serna, M.J.: Approximating layout problems on random geometric graphs. Journal of Algorithms 39, 78–117 (2001)
Díaz, J., Petit, J., Serna, M.J.: A survey of graph layout problems. ACM Computing Surveys 34, 313–356 (2002)
Gavril, F.: Some NP-complete problems on graphs. In: 11th Conference on Information Sciences and Systems, pp. 91–95. John Hopkins University, Baltimore (1977)
Heggernes, P., Lokshtanov, D., Mihai, R., Papadopoulos, C.: Cutwidth of split graphs, threshold graphs, and proper interval graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 218–229. Springer, Heidelberg (2008)
Jünger, M., Reinelt, G., Rinaldi, G.: The traveling salesman problem. In: Handbook on Operations Research and Management Sciences, vol. 7, pp. 225–330. North-Holland, Amsterdam (1995)
Karger, D.R.: A randomized fully polynomial approximation scheme for the all terminal network reliability problem. In: Proceedings of STOC 1996, pp. 11–17. ACM, New York (1996)
Leighton, F.T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: Proceedings of FOCS 1988, pp. 422–431. IEEE, Los Alamitos (1988)
Makedon, F., Sudborough, I.H.: Minimizing width in linear layouts. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 478–490. Springer, Heidelberg (1983)
Monien, B., Sudborough, I.H.: Min cut is NP-complete for edge weighted trees. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 265–274. Springer, Heidelberg (1986)
Mutzel, P.: A polyhedral approach to planar augmentation and related problems. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 497–507. Springer, Heidelberg (1995)
Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Applied Mathematics 18, 279–292 (1987)
Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth I: A linear time fixed parameter algorithm. Journal of Algorithms 56, 1–24 (2005)
Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth II: Algorithms for partial w-trees of bounded degree. Journal of Algorithms 56, 24–49 (2005)
Yannakakis, M.: A polynomial algorithm for the min cut linear arrangement of trees. Journal of ACM 32, 950–988 (1985)
Yuan, J., Zhou, S.: Optimal labelling of unit interval graphs. Appl. Math. J. Chinese Univ. Ser. B (English edition) 10, 337–344 (1995)
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Heggernes, P., van ’t Hof, P., Lokshtanov, D., Nederlof, J. (2010). Computing the Cutwidth of Bipartite Permutation Graphs in Linear Time. In: Thilikos, D.M. (eds) Graph Theoretic Concepts in Computer Science. WG 2010. Lecture Notes in Computer Science, vol 6410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16926-7_9
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