Quick Separation in Chordal and Split Graphs

Authors Pranabendu Misra, Fahad Panolan, Ashutosh Rai , Saket Saurabh, Roohani Sharma

Thumbnail PDF


  • Filesize: 0.63 MB
  • 14 pages

Document Identifiers

Author Details

Pranabendu Misra
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Fahad Panolan
  • Department of Computer Science and Engineering, IIT Hyderabad, India
Ashutosh Rai
  • Depaertment of Applied Mathematics, Charles University, Prague, Czech Republic
Saket Saurabh
  • Institute of Mathematical Sciences, HBNI, India
  • UMI ReLax, Chennai, India
  • University of Bergen, Norway
Roohani Sharma
  • Institute of Mathematical Sciences, HBNI, India

Cite AsGet BibTex

Pranabendu Misra, Fahad Panolan, Ashutosh Rai, Saket Saurabh, and Roohani Sharma. Quick Separation in Chordal and Split Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of 𝓁 vertex pairs (s_i, t_i), i ∈ [𝓁], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V(G)⧵ {s_i,t_i : i ∈ [𝓁]} of size at most k such that for every vertex pair (s_i,t_i), s_i and t_i are in two different connected components of G-S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(s_i,t_i): i ∈ [𝓁]}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T× T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2^{{𝒪}(k³)}n^{{𝒪}(1)} and 2^k n^{{𝒪}(1)}, respectively. However, the kernelization complexity of both these problems is not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results. 1) Multicut on chordal graphs admits a polynomial kernel with {𝒪}(k³ 𝓁⁷) vertices. Multiway Cut on chordal graphs admits a polynomial kernel with {𝒪}(k^{13}) vertices. 2) Multicut on chordal graphs can be solved in time min {𝒪(2^{k} ⋅ (k³+𝓁) ⋅ (n+m)), 2^{𝒪(𝓁 log k)} ⋅ (n+m) + 𝓁 (n+m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP. 3) Multicut on split graphs can be solved in time min {𝒪(1.2738^k + kn+𝓁(n+m), 𝒪(2^{𝓁} ⋅ 𝓁 ⋅ (n+m))}. Unrestricted Multicut on split graphs can be solved in time 𝒪(4^{𝓁}⋅ 𝓁 ⋅ (n+m)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • chordal graphs
  • multicut
  • multiway cut
  • FPT
  • kernel


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nicolas Bousquet, Jean Daligault, and Stéphan Thomassé. Multicut is FPT. SIAM J. Comput., 47(1):166-207, 2018. Google Scholar
  2. Jianer Chen, Iyad A Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(40-42):3736-3756, 2010. Google Scholar
  3. Jianer Chen, Yang Liu, and Songjian Lu. An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica, 55(1):1-13, 2009. Google Scholar
  4. Marek Cygan, Stefan Kratsch, Marcin Pilipczuk, Michał Pilipczuk, and Magnus Wahlström. Clique cover and graph separation: New incompressibility results. TOCT, 6(2):6:1-6:19, 2014. Google Scholar
  5. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. TOCT, 5(1):3:1-3:11, 2013. Google Scholar
  6. Gruia Călinescu, Cristina G. Fernandes, and Bruce Reed. Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width. Journal of Algorithms, 48(2):333-359, 2003. Google Scholar
  7. E. Dahlhaus, D. Johnson, C. Papadimitriou, P. Seymour, and M. Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. Google Scholar
  8. Elias Dahlhaus. Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition. J. Algorithms, 36(2):205-240, 2000. Google Scholar
  9. Reinhard Diestel. Graph theory. 2005. Grad. Texts in Math, 101, 2005. Google Scholar
  10. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  11. Philippe Galinier, Michel Habib, and Christophe Paul. Chordal graphs and their clique graphs. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science, 21st International Workshop, WG '95, Aachen, Germany, June 20-22, 1995, Proceedings, volume 1017 of Lecture Notes in Computer Science, pages 358-371. Springer, 1995. Google Scholar
  12. M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980. Google Scholar
  13. Jiong Guo, Falk Hüffner, Erhan Kenar, Rolf Niedermeier, and Johannes Uhlmann. Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs. European Journal of Operational Research, 186(2):542-553, 2008. Google Scholar
  14. Bart M. P. Jansen, Marcin Pilipczuk, and Erik Jan van Leeuwen. A Deterministic Polynomial Kernel for Odd Cycle Transversal and Vertex Multiway Cut in Planar Graphs. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), volume 126 of Leibniz International Proceedings in Informatics (LIPIcs), pages 39:1-39:18, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  15. Philip N. Klein and Dániel Marx. Solving planar k -terminal cut in o(n^c√k) time. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 569-580. Springer, 2012. Google Scholar
  16. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 450-459. IEEE Computer Society, 2012. Google Scholar
  17. Dániel Marx. Parameterized graph separation problems. Theor. Comput. Sci., 351(3):394-406, 2006. Google Scholar
  18. Dániel Marx and Igor Razgon. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput., 43(2):355-388, 2014. Google Scholar
  19. Charis Papadopoulos. Restricted vertex multicut on permutation graphs. Discrete Applied Mathematics, 160(12):1791-1797, 2012. Google Scholar