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Enumeration of minimal transversals of hypergraphs of bounded VC-dimension

Arnaud Mary

TL;DR

This work addresses the transversal hypergraph problem by proving that Trans-Hyp (and thus Trans-Enum) is solvable in polynomial time when one input hypergraph has bounded VC-dimension, using traces, k-traces, and the k-extension ext_k(H). The approach combines checking non-membership of certain k-traces with polynomial-time construction of ext_k(H), leveraging Sauer–Shelah bounds to keep growth in check, and yields quasi-polynomial behavior in general hypergraphs. It unifies and extends many known polynomial cases by showing polynomiality for classes closed under partial subhypergraphs and for k-conformal hypergraphs, while highlighting open questions about FPT status and induced-subhypergraph restrictions. Overall, the results significantly broaden tractable instances of hypergraph dualization and provide a concrete algorithmic framework for incremental enumeration in bounded VC-dimension settings.

Abstract

We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.

Enumeration of minimal transversals of hypergraphs of bounded VC-dimension

TL;DR

This work addresses the transversal hypergraph problem by proving that Trans-Hyp (and thus Trans-Enum) is solvable in polynomial time when one input hypergraph has bounded VC-dimension, using traces, k-traces, and the k-extension ext_k(H). The approach combines checking non-membership of certain k-traces with polynomial-time construction of ext_k(H), leveraging Sauer–Shelah bounds to keep growth in check, and yields quasi-polynomial behavior in general hypergraphs. It unifies and extends many known polynomial cases by showing polynomiality for classes closed under partial subhypergraphs and for k-conformal hypergraphs, while highlighting open questions about FPT status and induced-subhypergraph restrictions. Overall, the results significantly broaden tractable instances of hypergraph dualization and provide a concrete algorithmic framework for incremental enumeration in bounded VC-dimension settings.

Abstract

We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph . An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.
Paper Structure (11 sections, 23 theorems, 2 equations, 1 algorithm)

This paper contains 11 sections, 23 theorems, 2 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. Main algorithm
  5. Finding a minimal transversal satisfying a $k$-trace
  6. Computing $ext_{k}(\mathcal{H})$
  7. Consequences
  8. Hypergraph classes closed under subhypergraphs
  9. Classes closed under partial subhypergraphs.
  10. Other subhypergraphs types
  11. Discussions

Key Result

Theorem 0

Let $\mathcal{H}$ be a hypergraph with $\textsc{VC-dim}\xspace(\mathcal{H})<k$ and let $\mathcal{G} \subseteq Tr(\mathcal{H})$. One can decide in time $O(2^k(n|\mathcal{G}|)^{k+1} + n^{2k+1}|\mathcal{G}|)$ whether $\mathcal{G} = Tr(\mathcal{H})$ and find $T\in Tr(\mathcal{H}) \setminus \mathcal{G}$

Theorems & Definitions (39)

  • Theorem 0
  • Corollary 0
  • Lemma 1: Sauer–Shelah Lemma
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4: boros_dual_1998
  • Lemma 5
  • proof
  • ...and 29 more