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

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

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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

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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}.

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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

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

Funding

  • Agrawal, Akanksha: Supported by the PBC Program of Fellowships for Outstanding Post-doctoral Researchers from China and India (No. 5101479000).
  • Saurabh, Saket: Supported by the Horizon 2020 Framework program, ERC Consolidator Grant LOPPRE (No. 819416).

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