Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices

Authors Akanksha Agrawal, Sudeshna Kolay, Jayakrishnan Madathil, Saket Saurabh



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2019.41.pdf
  • Filesize: 2.8 MB
  • 14 pages

Document Identifiers

Author Details

Akanksha Agrawal
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Sudeshna Kolay
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Jayakrishnan Madathil
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Saket Saurabh
  • University of Bergen, Bergen, Norway
  • The Institute of Mathematical Sciences, HBNI, Chennai, India

Cite As Get BibTex

Akanksha Agrawal, Sudeshna Kolay, Jayakrishnan Madathil, and Saket Saurabh. Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.41

Abstract

Given a symmetric l x l matrix M=(m_{i,j}) with entries in {0,1,*}, a graph G and a function L : V(G) - > 2^{[l]} (where [l] = {1,2,...,l}), a list M-partition of G with respect to L is a partition of V(G) into l parts, say, V_1, V_2, ..., V_l such that for each i,j in {1,2,...,l}, (i) if m_{i,j}=0 then for any u in V_i and v in V_j, uv not in E(G), (ii) if m_{i,j}=1 then for any (distinct) u in V_i and v in V_j, uv in E(G), (iii) for each v in V(G), if v in V_i then i in L(v). We consider the Deletion to List M-Partition problem that takes as input a graph G, a list function L:V(G) - > 2^[l] and a positive integer k. The aim is to determine whether there is a k-sized set S subseteq V(G) such that G-S has a list M-partition. Many important problems like Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, Multiway Cut and Deletion to List Homomorphism are special cases of the Deletion to List M-Partition problem. In this paper, we provide a classification of the parameterized complexity of Deletion to List M-Partition, parameterized by k, (a) when M is of order at most 3, and (b) when M is of order 4 with all diagonal entries belonging to {0,1}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • list matrix partitions
  • parameterized classification
  • Almost 2-SAT
  • important separators
  • iterative compression

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Noga Alon and Michael Tarsi. Colorings and orientations of graphs. Combinatorica, 12(2):125-134, 1992. Google Scholar
  2. Bengt Aspvall, Michael F. Plass, and Robert E. Tarjan. A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas. Information Processing Letters, 8(3):121-123, 1979. Google Scholar
  3. Armen S. Asratian, Tristan M. J. Denley, and Roland Häggkvist. Bipartite graphs and their applications, volume 131. Cambridge University Press, 1998. Google Scholar
  4. Andreas Brandstädt, Feodor F. Dragan, Van Bang Le, and Thomas Szymczak. On stable cutsets in graphs. Discrete Applied Mathematics, 105(1-3):39-50, 2000. URL: https://doi.org/10.1016/S0166-218X(00)00197-9.
  5. Kathie Cameron, Elaine M. Eschen, Chính T. Hoàng, and R. Sritharan. The Complexity of the List Partition Problem for Graphs. SIAM J. Discrete Math., 21(4):900-929, 2007. Google Scholar
  6. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM, 55(5):21:1-21:19, 2008. Google Scholar
  7. Rajesh Chitnis, László Egri, and Dániel Marx. List H-Coloring a Graph by Removing Few Vertices. Algorithmica, 78(1):110-146, 2017. Google Scholar
  8. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socała. Tight Bounds for Graph Homomorphism and Subgraph Isomorphism. In ACM-SIAM Symposium on Discrete Algorithms, pages 1643-1649, 2016. Google Scholar
  9. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The Complexity of Multiway Cuts (Extended Abstract). In ACM Symposium on Theory of Computing (STOC), pages 241-251, 1992. Google Scholar
  10. Josep Díaz, Maria Serna, and Dimitrios M. Thilikos. (H,C,K)-coloring: Fast, easy, and hard cases. In Mathematical Foundations of Computer Science (MFCS), pages 304-315, 2001. Google Scholar
  11. Josep Díaz, Maria Serna, and Dimitrios M Thilikos. Recent results on parameterized H-colorings. Discrete Mathematics and Theoretical Computer Science, 63:65-86, 2004. Google Scholar
  12. László Egri, Andrei Krokhin, Benoit Larose, and Pascal Tesson. The complexity of the list homomorphism problem for graphs. Theory of Computing Systems, 51(2):143-178, 2012. Google Scholar
  13. Paul Erdős, Arthur L. Rubin, and Herbert Taylor. Choosability in graphs. In Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium, volume 26, pages 125-157, 1979. Google Scholar
  14. Tomás Feder and Pavol Hell. List Homomorphisms to Reflexive Graphs. Journal of Combinatorial Theory, Series B, 72(2):236-250, 1998. Google Scholar
  15. Tomás Feder and Pavol Hell. Full Constraint Satisfaction Problems. SIAM J. Comput., 36(1):230-246, 2006. URL: https://doi.org/10.1137/S0097539703427197.
  16. Tomás Feder, Pavol Hell, and Jing Huang. List Homomorphisms and Circular Arc Graphs. Combinatorica, 19(4):487-505, October 1999. Google Scholar
  17. Tomás Feder, Pavol Hell, Sulamita Klein, and Rajeev Motwani. List Partitions. Journal of Discrete Mathematics, 16(3):449-478, 2003. Google Scholar
  18. Lester R. Ford and Delbert R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8(3):399-404, 1956. Google Scholar
  19. Michael R. Garey and David S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. Computers and Intractability, page 340, 1979. Google Scholar
  20. Michael. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some Simplified NP-Complete Graph Problems. Theor. Comput. Sci., 1(3):237-267, 1976. URL: https://doi.org/10.1016/0304-3975(76)90059-1.
  21. Andreas Göbel, Leslie Ann Goldberg, Colin McQuillan, David Richerby, and Tomoyuki Yamakami. Counting List Matrix Partitions of Graphs. SIAM J. Comput., 44(4):1089-1118, 2015. URL: https://doi.org/10.1137/140963029.
  22. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Elsevier, 2004. Google Scholar
  23. Melven R. Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Mathematical Logic Quarterly, 13(1-2):15-20, 1967. Google Scholar
  24. Marek Kubale. Some results concerning the complexity of restricted colorings of graphs. Discrete Applied Mathematics, 36(1):35-46, 1992. Google Scholar
  25. S. Føldes and P. L. Hammer. Split graphs. South-Eastern Conference on Combinatorics, Graph Theory and Computing (SEICCGTC), pages 311-315, 1977. Google Scholar
  26. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster Parameterized Algorithms Using Linear Programming. Transactions on Algorithms, 11(2):15:1-15:31, 2014. Google Scholar
  27. Dániel Marx. Parameterized graph separation problems. Theoretical Computer Science, 351(3):394-406, 2006. Google Scholar
  28. Vadim G. Vizing. Vertex colorings with given colors. Diskret. Analiz, 29:3-10, 1976. Google Scholar
  29. Margit Voigt. List colourings of planar graphs. Discrete Mathematics, 120(1-3):215-219, 1993. Google Scholar
  30. Mihalis Yannakakis. Node-and Edge-deletion NP-complete Problems. In ACM Symposium on Theory of Computing (STOC), pages 253-264, 1978. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail