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Enumeration of Minimal Hitting Sets Parameterized by Treewidth

Batya Kenig, Dan Shlomo Mizrahi

TL;DR

The paper tackles the problem of enumerating minimal hitting sets (transversals) of a hypergraph parameterized by treewidth. It reduces trans-enum to dom-enum via a tripartite augmentation that preserves bounded treewidth, and develops a dedicated enumeration algorithm for minimal dominating sets based on disjoint-branch tree decompositions. The main result is a fixed-parameter-linear-delay enumeration with preprocessing time $O((n+m)k5^k)$ and memory $O((n+m)5^k$ when $w \le k \le 2w$, representing a significant improvement over prior $O(||\mathcal{H}|| \cdot n^{w+1})$ delay bounds. This approach has practical impact for data-management tasks such as discovering UCCs, generating database repairs, and related Trans-Enum applications in bounded-treewidth settings.

Abstract

Enumerating the minimal hitting sets of a hypergraph is a problem which arises in many data management applications that include constraint mining, discovering unique column combinations, and enumerating database repairs. Previously, Eiter et al. showed that the minimal hitting sets of an $n$-vertex hypergraph, with treewidth $w$, can be enumerated with delay $O^*(n^{w})$ (ignoring polynomial factors), with space requirements that scale with the output size. We improve this to fixed-parameter-linear delay, following an FPT preprocessing phase. The memory consumption of our algorithm is exponential with respect to the treewidth of the hypergraph.

Enumeration of Minimal Hitting Sets Parameterized by Treewidth

TL;DR

The paper tackles the problem of enumerating minimal hitting sets (transversals) of a hypergraph parameterized by treewidth. It reduces trans-enum to dom-enum via a tripartite augmentation that preserves bounded treewidth, and develops a dedicated enumeration algorithm for minimal dominating sets based on disjoint-branch tree decompositions. The main result is a fixed-parameter-linear-delay enumeration with preprocessing time and memory when , representing a significant improvement over prior delay bounds. This approach has practical impact for data-management tasks such as discovering UCCs, generating database repairs, and related Trans-Enum applications in bounded-treewidth settings.

Abstract

Enumerating the minimal hitting sets of a hypergraph is a problem which arises in many data management applications that include constraint mining, discovering unique column combinations, and enumerating database repairs. Previously, Eiter et al. showed that the minimal hitting sets of an -vertex hypergraph, with treewidth , can be enumerated with delay (ignoring polynomial factors), with space requirements that scale with the output size. We improve this to fixed-parameter-linear delay, following an FPT preprocessing phase. The memory consumption of our algorithm is exponential with respect to the treewidth of the hypergraph.
Paper Structure (4 sections, 9 theorems, 1 equation, 1 figure)

This paper contains 4 sections, 9 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. From Trans-Enum to Dom-Enum
  4. Overview of Algorithm

Key Result

Theorem 1

Let $G$ be an $n$-vertex graph whose treewidth is $w$. Following a preprocessing phase that takes time $O(nw8^{2w})$, the minimal dominating sets of $G$ can be enumerated with delay $O(nw)$ and total space $O(n8^{2w})$.

Theorems & Definitions (19)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • Definition 6: Tree Decomposition (TD)
  • Definition 7
  • Definition 8
  • Theorem 9
  • Lemma 10
  • ...and 9 more