Expansion Lemma—Variations and Applications to Polynomial-Time Preprocessing

Ashwin Jacob, Diptapriyo Majumdar, Venkatesh Raman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In parameterized complexity, it is well-known that a parameterized problem is fixed-parameter tractable if and only if it has a kernel—an instance equivalent to the input instance, whose size is just a function of the parameter. The size of the kernel can be exponential or worse, resulting in a quest for fixed-parameter tractable problems with polynomial-sized kernels. The developments in machinery (showing lower bounds for the sizes of the kernels) have led researchers to question what are the asymptotically optimum sizes for the kernels of fixed-parameter tractable problems. In this article, we surveyed a tool called expansion lemma that helps in reducing the size of the kernel. Its early origin was in the form of crown decomposition, i.e., to obtain the linear kernel for the Vertex Cover problem; the specific lemma was identified as the tool behind the optimal (Formula presented.) kernel for the undirected feedback vertex set problem. Since then, several variations and extensions of the tool have been discovered. We surveyed them along with their applications in this article.

Original languageEnglish
Article number144
Issue number3
StatePublished - 1 Mar 2023


  • crown decomposition
  • expansion lemma
  • kernelization
  • parameterized algorithms

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Numerical Analysis
  • Computational Theory and Mathematics
  • Computational Mathematics


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