Deconstructing Parameterized Hardness of Fair Vertex Deletion Problems

Ashwin Jacob, Venkatesh Raman, Vibha Sahlot

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations


For a graph G and a positive integer d, a set S ⊆ V (G) is a fair set with the fairness factor d if for every vertex in G, at most d of its neighbors are in the S. In the Π -Vertex Deletion problem, the aim is to find in a given graph a set S of minimum size such that G\S satisfies the property Π. We look at Π -Fair Vertex Deletion problem where we also want S to be a fair set. It is known that the general Π -Fair Vertex Deletion problem where Π is expressible by a first order formula and is also given as input is W[1]-hard when parameterized by treewidth. Our first observation is that if we also parameterize by the fairness constant d, then Π -Fair Vertex Deletion is FPT (fixed-parameter tractable) even if Π -Vertex Deletion can be expressed by a Monadic Second Order (MSO) formula. As a corollary we get an FPT algorithm for Fair Feedback Vertex Set (FFVS) and Fair Vertex Cover (FVC) parameterized by solution size. We then do a deep dive on FVC and more generally Π -Fair Vertex Deletion problems parameterized by solution size k when Π is characterized by a finite set of forbidden graphs. We show that these problems are FPT and develop a polynomial kernel when d is a constant. While the FPT algorithms use the standard branching technique, the fairness constraint introduces challenges to design a polynomial kernel. En route, we give a polynomial kernel for a special instance of Min-ones-SAT and Fair q-hitting set, a generalization of q-hitting set, with a fairness constraint on an underlying graph structure on the universe. These could be of independent interest. To complement our FPT results, we show that Fair Set and Fair Independent Set problems are W[1]-hard even in 3-degenerate graphs when the fairness factor is 1. We also show that FVC is polynomial-time solvable when d = 1 or 2, and NP -hard for d ≥ 3, and that FFVS is NP -hard for all d ≥ 1.

Original languageEnglish
Title of host publicationComputing and Combinatorics - 25th International Conference, COCOON 2019, Proceedings
EditorsDing-Zhu Du, Zhenhua Duan, Cong Tian
PublisherSpringer Verlag
Number of pages13
ISBN (Print)9783030261757
StatePublished - 1 Jan 2019
Externally publishedYes
Event25th International Computing and Combinatorics Conference, COCOON 2019 - Xi'an, China
Duration: 29 Jul 201931 Jul 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11653 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference25th International Computing and Combinatorics Conference, COCOON 2019


  • Kernelization
  • Parameterized complexity
  • Vertex deletion problems
  • W-hardness

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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