Exploiting Dense Structures in Parameterized Complexity

Authors William Lochet, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi

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Author Details

William Lochet
  • Department of Informatics, University of Bergen, Norway
Daniel Lokshtanov
  • University of California Santa Barbara, CA, USA
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • Department of Informatics, University of Bergen, Norway
Meirav Zehavi
  • Ben Gurion University of the Negev, Beer Sheva, Israel


We thank one of the referee of a different version of this paper for sketching a proof based on the random walk on expanders for Lemma 10.

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William Lochet, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Exploiting Dense Structures in Parameterized Complexity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most Ω(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Dense graphs
  • disjoint paths
  • odd cycle transversal
  • kernels


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