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Incompressibility of H-Free Edge Modification Problems: Towards a Dichotomy

Authors Dániel Marx, R. B. Sandeep

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Author Details

Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
R. B. Sandeep
  • Department of Computer Science and Engineering, Indian Institute of Technology Dharwad, India

Funding

Research supported by the European Research Council (ERC) grant SYSTEMATICGRAPH: "Systematic mapping of the complexity landscape of hard algorithmic graph problems", reference 725978.
  • Sandeep, R. B.: Partially supported by SERB Grant SRG 002276: "Complexity Dichotomies for Graph Modification Problems".

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Dániel Marx and R. B. Sandeep. Incompressibility of H-Free Edge Modification Problems: Towards a Dichotomy. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 72:1-72:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.72

Abstract

Given a graph G and an integer k, the H-free Edge Editing problem is to find whether there exist at most k pairs of vertices in G such that changing the adjacency of the pairs in G results in a graph without any induced copy of H. The existence of polynomial kernels for H-free Edge Editing (that is, whether it is possible to reduce the size of the instance to k^O(1) in polynomial time) received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs H with at most 4 vertices (e.g., path on 3 or 4 vertices, diamond, paw), but starting from 5 vertices, polynomial kernels are known only if H is either complete or empty. This suggests the conjecture that there is no other H with at least 5 vertices were H-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set ℋ of nine 5-vertex graphs such that if for every H ∈ ℋ, H-free Edge Editing is incompressible and the complexity assumption NP ⊈ coNP/poly holds, then H-free Edge Editing is incompressible for every graph H with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of H-free Edge Editing for every H with at least 5 vertices.
We obtain similar result also for H-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs H where the problem is trivial (graphs with exactly one edge). We obtain a larger set ℋ of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of H-free Edge Deletion for every graph H with at least 5 vertices. Analogous results follow also for the H-free Edge Completion problem by simple complementation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • incompressibility
  • edge modification problems
  • H-free graphs

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References

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