Abstract
Given is a connected positively weighted undirected planar graph G embedded in the plane, a source vertex s, and a set of sink vertices T. An (s,T)-cut in G corresponds to a cycle or a collection of edge-disjoint cycles in the planar dual graph G * that define a planar region containing s but not T. A cut with a connectivity prior does not separate the vertices in T from each other: we focus on the most natural prior where the cut corresponds to a (simple, i.e., no repeated vertices) cycle in G *. We present an algorithm that finds a minimum simple (s,T)-cut in O(n 4) time for n vertices. To the best of our knowledge, this is the first polynomial-time algorithm for minimum cuts with connectivity priors. Such cuts have applications in computer vision and medical imaging.
This material is based upon work supported by the National Science Foundation, Award No. CCF-1319987. Part of the work was done while the first author visited the Simons Institute for the Theory of Computing at the University of California, Berkeley.
This is a preview of subscription content, log in via an institution to check access.
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
Bateni, M., Hajiaghayi, M., Klein, P.N., Mathieu, C.: A polynomial-time approximation scheme for planar multiway cut. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). pp. 639–655 (2012)
Bezáková, I., Langley, Z.: Contiguous minimum single-source-multi-sink cuts in weighted planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 49–60. Springer, Heidelberg (2012)
Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1), 93–109 (1990)
Borradaile, G., Klein, P.N.: An O(nlogn) algorithm for maximum st-flow in a directed planar graph. J. ACM 56(2) (2009)
Borradaile, G., Klein, P.N., Mozes, S., Nussbaum, Y., Wulff-Nilsen, C.: Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. In: Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 170–179 (2011)
Boykov, Y., Veksler, O.: Graph cuts in vision and graphics: Theories and applications. In: Handbook of Mathematical Models in Computer Vision. Springer (2006)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)
Cabello, S.: Finding shortest contractible and shortest separating cycles in embedded graphs. ACM Trans. on Algorithms 6(2) (2010); Ext. abstr. in SODA 2009
Chalermsook, P., Fakcharoenphol, J., Nanongkai, D.: A deterministic near-linear time algorithm for finding minimum cuts in planar graphs. In: Proceedings of the 15th Annual ACM-SIAM Symp. on Discr. Algorithms (SODA), pp. 828–829 (2004)
Chambers, E.W., Erickson, J., Nayyeri, A.: Minimum cuts and shortest homologous cycles. In: Proceedings of the 25th Annual ACM Symposium on Computational Geometry (SCG), pp. 377–385 (2009)
Chambers, E.W., de Verdière, É.C., Erickson, J., Lazarus, F., Whittlesey, K.: Splitting (complicated) surfaces is hard. Comput. Geom. 41(1-2), 94–110 (2008)
Łącki, J., Sankowski, P.: Min-cuts and shortest cycles in planar graphs in O(n loglogn) time. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 155–166. Springer, Heidelberg (2011)
Vicente, S., Kolmogorov, V., Rother, C.: Graph cut based image segmentation with connectivity priors. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR (2008)
Zeng, Y., Samaras, D., Chen, W., Peng, Q.: Topology cuts: A novel min-cut/max-flow algorithm for topology preserving segmentation in N-D images. Computer Vision Image Understanding 112, 81–90 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bezáková, I., Langley, Z. (2014). Minimum Planar Multi-sink Cuts with Connectivity Priors. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44465-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-662-44465-8_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44464-1
Online ISBN: 978-3-662-44465-8
eBook Packages: Computer ScienceComputer Science (R0)