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Hypergraph dualization with FPT-delay parameterized by the degeneracy and dimension

Valentin Bartier, Oscar Defrain, Fionn Mc Inerney

TL;DR

The paper addresses the problem of enumerating minimal transversals (hypergraph dualization) with a delay that is fixed-parameter tractable with respect to degeneracy and dimension. It extends the ordered generation technique to hypergraphs, showing that the extension (children) of a partial transversal can be enumerated in time $k^{d}\cdot n^{O(1)}$ when the weak degeneracy is $d$ and the dimension is $k$, yielding an $FPT$-delay algorithm for Trans-Enum parameterized by these two parameters. As a corollary, minimal dominating sets in graphs can be enumerated within $(\Delta+1)^{d+1}\cdot n^{O(1)}$ time, tying domination complexity to hypergraph degeneracy and maximum degree. The results illuminate the tractability frontier for enumeration problems under degeneracy-based parameters, while also outlining limitations and connections to related tasks such as maximal irredundant sets and Multicolored Independent Set.

Abstract

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kanté, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay for a weaker notion of degeneracy, or even parameterized by the maximum degree $Δ$, i.e., whether it can be turned into an algorithm with delay $f(Δ)\cdot n^{O(1)}$ for some computable function $f$. Moreover, and as a first step toward answering that question, they note that they could not achieve these time bounds even for the particular case of minimal dominating sets enumeration. In this paper, using ordered generation, we show that an FPT-delay algorithm can be devised for minimal transversals enumeration parameterized by the degeneracy and dimension, giving a positive and more general answer to the latter question.

Hypergraph dualization with FPT-delay parameterized by the degeneracy and dimension

TL;DR

The paper addresses the problem of enumerating minimal transversals (hypergraph dualization) with a delay that is fixed-parameter tractable with respect to degeneracy and dimension. It extends the ordered generation technique to hypergraphs, showing that the extension (children) of a partial transversal can be enumerated in time when the weak degeneracy is and the dimension is , yielding an -delay algorithm for Trans-Enum parameterized by these two parameters. As a corollary, minimal dominating sets in graphs can be enumerated within time, tying domination complexity to hypergraph degeneracy and maximum degree. The results illuminate the tractability frontier for enumeration problems under degeneracy-based parameters, while also outlining limitations and connections to related tasks such as maximal irredundant sets and Multicolored Independent Set.

Abstract

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an -delay algorithm listing all minimal transversals of an -vertex hypergraph of degeneracy . Recently at IWOCA 2019, Conte, Kanté, Marino, and Uno asked whether this XP-delay algorithm parameterized by could be made FPT-delay for a weaker notion of degeneracy, or even parameterized by the maximum degree , i.e., whether it can be turned into an algorithm with delay for some computable function . Moreover, and as a first step toward answering that question, they note that they could not achieve these time bounds even for the particular case of minimal dominating sets enumeration. In this paper, using ordered generation, we show that an FPT-delay algorithm can be devised for minimal transversals enumeration parameterized by the degeneracy and dimension, giving a positive and more general answer to the latter question.
Paper Structure (10 sections, 8 theorems, 4 equations, 2 figures)

This paper contains 10 sections, 8 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. Graphs and hypergraphs.
  5. Graph domination.
  6. Ordered generation
  7. Parameterized children generation
  8. Consequences for minimal domination
  9. Discussion and limitations
  10. Acknowledgements.

Key Result

Theorem 1.1

There is an algorithm that, given an $n$-vertex hypergraph of weak degeneracy $d$, generates its minimal transversals with $n^{O(d)}$-delay.

Figures (2)

  • Figure 1: A hypergraph $\mathcal{H}$ on $6$ vertices and $3$ edges. The hypergraph induced by $S:=\{v_1,v_2,v_3,v_4\}$ has a single edge $\{v_1,v_4\}$, while the trace of $S$ on $\mathcal{H}$ contains three edges $\{v_1,v_4\}$, $\{v_2,v_4\}$, and $\{v_3,v_4\}$.
  • Figure 2: A representation of the solution tree defined by the $\mathop{\mathrm{\sf parent}}\nolimits$ relation.

Theorems & Definitions (18)

  • Theorem 1.1: eiter2002neweiter2003new
  • Theorem 1.5
  • Definition 1: Weak degeneracy, eiter2003new
  • Definition 2: Strong degeneracy, kostochka2008adaptedconte2019maximal
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 8 more