Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Descriptive Complexity of Vertex Deletion Problems

Max Bannach, Florian Chudigiewitsch, Till Tantau

TL;DR

This paper refined this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern.

Abstract

Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property. Regarding k as the parameter, a dichotomy was recently shown based on the number of quantifier alternations of first-order formulas that describe the property. In this paper, we refine this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern. The more fine-grained approach uncovers new tractable fragments, which we show to not only lie in FPT, but even in parameterized constant-depth circuit complexity classes. On the other hand, we show that vertex deletion becomes intractable already for just one quantifier per alternation, that is, there is a formula of the form {\forall}x{\exists}y{\forall}z(ψ), with ψ quantifier-free, for which the vertex deletion problem is W[1]-hard. The fine-grained analysis also allows us to uncover differences in the complexity landscape when we consider different kinds of graphs and more general structures: While basic graphs (undirected graphs without self-loops), undirected graphs, and directed graphs each have a different frontier of tractability, the frontier for arbitrary logical structures coincides with that of directed graphs.

On the Descriptive Complexity of Vertex Deletion Problems

TL;DR

This paper refined this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern.

Abstract

Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property. Regarding k as the parameter, a dichotomy was recently shown based on the number of quantifier alternations of first-order formulas that describe the property. In this paper, we refine this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern. The more fine-grained approach uncovers new tractable fragments, which we show to not only lie in FPT, but even in parameterized constant-depth circuit complexity classes. On the other hand, we show that vertex deletion becomes intractable already for just one quantifier per alternation, that is, there is a formula of the form {\forall}x{\exists}y{\forall}z(ψ), with ψ quantifier-free, for which the vertex deletion problem is W[1]-hard. The fine-grained analysis also allows us to uncover differences in the complexity landscape when we consider different kinds of graphs and more general structures: While basic graphs (undirected graphs without self-loops), undirected graphs, and directed graphs each have a different frontier of tractability, the frontier for arbitrary logical structures coincides with that of directed graphs.
Paper Structure (6 sections, 16 theorems, 1 table)

This paper contains 6 sections, 16 theorems, 1 table.

Table of Contents

  1. Introduction
  2. Background in Descriptive and Parameterized Complexity
  3. Basic Graphs
  4. Undirected Graphs
  5. Directed Graphs and Arbitrary Structures
  6. Conclusion

Key Result

Theorem 2

Let $p \in \{a,e\}^*$ be a pattern.

Theorems & Definitions (16)

  • Theorem 2: Complexity Trichotomy for $\mathrm{p\text{\normalfont-}} \text{\normalfont\small$\mathrm{VD}$}_{\mathrm{basic}}(p)$
  • Lemma 3: Detailed Bounds for $\mathrm{p\text{\normalfont-}} \text{\normalfont\small$\mathrm{VD}$}_{\mathrm{basic}}(p)$
  • Lemma 4
  • Lemma 7
  • Lemma 9
  • Lemma 12
  • Lemma 13
  • Lemma 14
  • Lemma 15
  • Theorem 19: Complexity Trichotomy for $\mathrm{p\text{\normalfont-}} \text{\normalfont\small$\mathrm{VD}$}_{\mathrm{undir}}(p)$
  • ...and 6 more