Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Marx, Dániel; Sandeep, R. B. https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
when quoting this document, please refer to the following
DOI:
URN: urn:nbn:de:0030-drops-129383
URL:

;

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

pdf-format:


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.

BibTeX - Entry

@InProceedings{marx_et_al:LIPIcs:2020:12938,
  author =	{D{\'a}niel Marx and R. B. Sandeep},
  title =	{{Incompressibility of H-Free Edge Modification Problems: Towards a Dichotomy}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{72:1--72:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12938},
  URN =		{urn:nbn:de:0030-drops-129383},
  doi =		{10.4230/LIPIcs.ESA.2020.72},
  annote =	{Keywords: incompressibility, edge modification problems, H-free graphs}
}

Keywords: incompressibility, edge modification problems, H-free graphs
Seminar: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue date: 2020
Date of publication: 26.08.2020


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI