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Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels

Authors Akanksha Agrawal, M. S. Ramanujan



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Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, India
M. S. Ramanujan
  • University of Warwick, Coventry, UK

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Akanksha Agrawal and M. S. Ramanujan. Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 36:1-36:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.36

Abstract

The problem of computing a minimum set of vertices intersecting a finite set of forbidden minors in a given graph is a fundamental graph problem in the area of kernelization with numerous well-studied special cases. A major breakthrough in this line of research was made by Fomin et al. [FOCS 2012], who showed that the ρ-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most ρ) has a polynomial kernel of size k^g(ρ) for some function g. A second standout result in this line is that of Giannapoulou et al. [ACM TALG 2017], who obtained an f(η)k^𝒪(1)-size kernel (for some function f) for the η-Treedepth Modulator problem (delete fewest number of vertices to make treedepth at most η) and showed that some dependence of the exponent of k on ρ in the result of Fomin et al. for the ρ-Treewidth Modulator problem is unavoidable under reasonable complexity hypotheses. In this work, we provide an approximate interpolation between these two results by giving, for every ε > 0, a (1+ε)-approximate kernel of size f'(η,ρ,1/ε)⋅ k^g'(ρ) (for some functions f' and g') for the problem of deciding whether k vertices can be deleted from a given graph to obtain a graph that has elimination distance at most η to the class of graphs that have treewidth at most ρ. Graphs of treedepth η are precisely the graphs with elimination distance at most η-1 to the graphs of treewidth 0 and graphs of treewidth ρ are simply graphs with elimination distance 0 to graphs of treewidth ρ. Consequently, our result "approximately" interpolates between these two major results in this active line of research.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Lossy Kernelization
  • Treewidth Modulator
  • Vertex Deletion Problems

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References

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