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MaxMin Separation Problems: FPT Algorithms for st-Separator and Odd Cycle Transversal

Authors Ajinkya Gaikwad , Hitendra Kumar, Soumen Maity, Saket Saurabh , Roohani Sharma

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Author Details

Ajinkya Gaikwad
  • Indian Institute of Science Education and Research, Pune, India
Hitendra Kumar
  • Indian Institute of Science Education and Research, Pune, India
Soumen Maity
  • Indian Institute of Science Education and Research, Pune, India
Saket Saurabh
  • The Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway
Roohani Sharma
  • University of Bergen, Norway

Funding

  • Gaikwad, Ajinkya: The author is supported by the Ministry of Human Resource Development, Government of India, under Prime Minister’s Research Fellowship Scheme (No. MRF-192002-211).
  • Saurabh, Saket: The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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Ajinkya Gaikwad, Hitendra Kumar, Soumen Maity, Saket Saurabh, and Roohani Sharma. MaxMin Separation Problems: FPT Algorithms for st-Separator and Odd Cycle Transversal. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.36

Abstract

In this paper, we study the parameterized complexity of the MaxMin versions of two fundamental separation problems: Maximum Minimal st-Separator and Maximum Minimal Odd Cycle Transversal (OCT), both parameterized by the solution size. In the Maximum Minimal st-Separator problem, given a graph G, two distinct vertices s and t and a positive integer k, the goal is to determine whether there exists a minimal st-separator in G of size at least k. Similarly, the Maximum Minimal OCT problem seeks to determine if there exists a minimal set of vertices whose deletion results in a bipartite graph, and whose size is at least k. We demonstrate that both problems are fixed-parameter tractable parameterized by k. Our FPT algorithm for Maximum Minimal st-Separator answers the open question by Hanaka, Bodlaender, van der Zanden & Ono [TCS 2019]. 
One unique insight from this work is the following. We use the meta-result of Lokshtanov, Ramanujan, Saurabh & Zehavi [ICALP 2018] that enables us to reduce our problems to highly unbreakable graphs. This is interesting, as an explicit use of the recursive understanding and randomized contractions framework of Chitnis, Cygan, Hajiaghayi, Pilipczuk & Pilipczuk [SICOMP 2016] to reduce to the highly unbreakable graphs setting (which is the result that Lokshtanov et al. tries to abstract out in their meta-theorem) does not seem obvious because certain "extension" variants of our problems are W[1]-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Parameterized Complexity
  • FPT
  • MaxMin problems
  • Maximum Minimal st-separator
  • Maximum Minimal Odd Cycle Transversal
  • Unbreakable Graphs
  • CMSO
  • Long Induced Odd Cycles
  • Sunflower Lemma

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