Enumeration of minimal transversals of hypergraphs of bounded VC-dimension

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.

💡 Research Summary

The paper addresses the long‑standing open problem of efficiently enumerating all minimal transversals (minimal hitting sets) of a hypergraph H, also known as the hypergraph dualization or transversal hypergraph problem. While the best known general algorithm runs in quasi‑polynomial time (Fredman–Khachiyan, 1996), it has been unclear whether a polynomial‑time solution exists. The authors introduce a new parameter, the Vapnik‑Chervonenkis (VC) dimension of a hypergraph, and prove that if at least one of the two hypergraphs involved in the decision version (Trans‑Hyp) has bounded VC‑dimension, the problem can be solved in polynomial time. Equivalently, for any hypergraph H with VC‑dim(H) < k (k a constant), there exists an incremental polynomial‑time algorithm that enumerates all minimal transversals of H.

Key technical contributions:

1. Trace‑Based Characterisation – The authors define a k‑trace as a pair (T,S) where S⊆V(H), |S|=k, and T=S∩E for some hyperedge E. A set E is k‑compatible with H if every k‑subset of vertices induces a trace already present in H. The k‑extension extₖ(H) consists of all k‑compatible subsets. Crucially, if VC‑dim(H) < k then extₖ(H) has size O(|V|ᵏ) (Sauer–Shelah Lemma), and can be constructed in O(n^{2k}) time.

2. Algorithm 1 (Dualize) – Given hypergraphs H (VC‑dim < k) and G⊆Tr(H), the algorithm decides whether G=Tr(H) or finds a new minimal transversal in Tr(H)\G. It proceeds in two loops:

   • Loop 1 enumerates all k‑traces missing from H (there are at most 2ᵏ·C(n,k) of them) and checks, for each, whether a minimal transversal of G realises that trace. This reduces to testing whether a bounded‑size vertex set T is a sub‑transversal of a sub‑hypergraph of G, a problem known to be solvable in O(n·|G|·|T|) time (Lemma 6, from previous work).
   • Loop 2 iterates over all sets E in extₖ(H) that are not contained in any hyperedge of H. For each such E it tests membership in Tr(G). If any E belongs to Tr(G), it is a required new minimal transversal.

   The overall running time is O(2ᵏ·(n·|G|)^{k+1} + n·2^{k+1}·|G|), which is polynomial when k is a constant.

3. Generalisation of Known Polynomial Cases – Many previously identified polynomial‑time classes (bounded edge size, bounded intersection, β‑acyclic, δ‑sparse, bounded degree, bounded tree‑width, totally unimodular, balanced hypergraphs, and various geometric hypergraphs) all have bounded VC‑dimension; thus they fall under the new theorem. The authors also show that any class closed under partial subhypergraphs inherits polynomial solvability (Corollary 2).

4. Beyond VC‑Dimension: Bounded Conformality – The algorithm remains polynomial for k‑conformal hypergraphs, a class where each vertex belongs to at most k maximal cliques (or equivalently, each hyperedge is the intersection of at most k maximal hyperedges). This is significant because k‑conformal hypergraphs can have unbounded VC‑dimension, providing one of the few known polynomial cases beyond VC‑bounded families.

5. Complexity Landscape – For arbitrary hypergraphs the algorithm runs in quasi‑polynomial time, matching the best known bound. For VC‑bounded hypergraphs it improves to incremental polynomial time, and for k‑conformal hypergraphs it again yields polynomial time.

The paper’s methodology combines combinatorial insights (trace theory, VC‑dimension, Sauer–Shelah bound) with algorithmic techniques (sub‑transversal testing, incremental construction of extₖ(H)). By focusing on the trace structure rather than the full hyperedge set, the authors avoid the exponential blow‑up that typically hampers dualization algorithms.

Implications:

• Theoretical: Provides a unifying parameter (VC‑dimension) that explains why many previously disparate polynomial cases exist, and settles an open question posed in earlier work (e.g.,