Near-linear time subhypergraph counting in bounded degeneracy hypergraphs

TL;DR

The work provides a tight dichotomy for counting small hypergraph patterns in bounded-degeneracy hypergraphs: a pattern $H$ is countable in near-linear time if it avoids a carefully defined family of obstructions $\\\\ H_{l}$ ITS; otherwise, under the Simplex Detection Conjecture, no $n^{1+\gamma}$-time algorithm exists. Central to the approach are directed acyclic hypergraphs, $l$-skeletons, and an $l$-degeneracy orientation that yield a DAG-tree decomposition with $\tau_{l}(H)$ guiding the exponent in the running time $\mathrm{poly}(\kappa_{l}) n^{\tau_{l}(H)} \log m$. The paper also extends the homomorphism counting framework to subhypergraphs via the quotient/homomorphism basis and tensor-product constructions, enabling a full dichotomy for Sub$_{H}(G)$ under the same obstructions. These results offer a principled, theory-backed foundation for near-linear time hypergraph pattern counting with potential database and network applications where hypergraph degeneracy is bounded. Key ingredients include the obstruction family $\\\\\\\\\ ext{\mathcal{H}}_{l}$, the notion of $\tau_{l}$, $l$-DAG-treewidth, and the reduction from general Hom counts to directed variants amenable to efficient counting. The framework unifies and extends prior graph-centric results to a broad hypergraph setting, including $l=0$ and $l=\infty$ cases, and clarifies when hypergraph pattern counting remains tractable.

Abstract

Counting small patterns in a large dataset is a fundamental algorithmic task. The most common version of this task is subgraph/homomorphism counting, wherein we count the number of occurrences of a small pattern graph $H$ in an input graph $G$. The study of this problem is a field in and of itself. Recently, both in theory and practice, there has been an interest in \emph{hypergraph} algorithms, where $G = (V,E)$ is a hypergraph. One can view $G$ as a set system where hyperedges are subsets of the universe $V$. Counting patterns $H$ in hypergraphs is less studied, although there are many applications in network science and database algorithms. Inspired by advances in the graph literature, we study when linear time algorithms are possible. We focus on input hypergraphs $G$ that have bounded \emph{degeneracy}, a well-studied concept for graph algorithms. We give a spectrum of definitions for hypergraph degeneracy that cover all existing notions. For each such definition, we give a precise characterization of the patterns $H$ that can be counted in (near) linear time. Specifically, we discover a set of ``obstruction patterns". If $H$ does not contain an obstruction, then the number of $H$-subhypergraphs can be counted exactly in $O(n\log n)$ time (where $n$ is the number of vertices in $G$). If $H$ contains an obstruction, then (assuming hypergraph variants of fine-grained complexity conjectures), there is a constant $γ> 0$, such that there is no $o(n^{1+γ})$ time algorithm for counting $H$-subhypergraphs. These sets of obstructions can be defined for all notions of hypergraph degeneracy.

Figures (7)

• Figure 1: An example of a graph $G$. Let $V$ be the set of black vertices. $G \langle V \rangle_{l}$ represents the $l$-trimmed subhypergraph of $G$ induced by the vertices in $V$. A hyperedge is only included in the subhypergraph when it has at most $l$ vertices outside $V$.
• Figure 2: Two examples of obstructions in $\mathcal{H}_{0}$. The example on the left has a core with $k=3$. Removing the vertices from the core disconnects the hypergraph into $3$ connected components (single vertices $D_1-D_3$). The sets $E_1-E_3$ correspond to the hyperedges containing each $D_i$. The example on the right has a core with $k=4$. Removing the core we end up with $5$ connected components. Note that we have an extra component $D_5$.
• Figure 3: Left: An obstruction in $\mathcal{H}_{0}$. Center: A pattern that contains the obstruction as a ($0$-trimmed) subhypergraph when induced by the black vertices, but it does not contain any obstruction as a $\infty$-trimmed subhypergraph. Therefore, it can be counted efficiently. Right: A pattern that contains the obstruction as the $\infty$-trimmed hypergraph induced by the black vertices. Note that the induced $0$-trimmed subhypergraph only contains the red hyperedge. This pattern cannot be counted efficiently.
• Figure 4: An example of an obstruction with a core of size $5$. Adding the gray hyperedge no longer results in an obstruction. Adding an external vertex to the gray hyperedge creates a new obstruction.
• Figure 5: More examples of obstructions.

Theorems & Definitions (83)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Conjecture 1.5: Simplex detection conjecture
• Theorem 1.6: name=Main Theorem
• Theorem 1.7: Characterization for subhypergraphs
• Claim 1.7
• Definition 2.1: Directed acyclic hypergraph
• Lemma 2.1