Stefano Clemente | Searcharxiv

arXiv2026

The Power of Bidegeneracy for Hypergraph Data Structures

A hypergraph $H = (V,E)$ describes a set system over a vertex universe $V$. Hypergraphs are increasingly popular in network science, as they represent interactions between subsets of vertices. A natural challenge is to design a data structure for $H$ to quickly answer basic questions about it, e.g. how many neighbors a vertex $v$ has or whether a hyperedge contains a vertex subset $S$. Such questions are trivial for graphs, for which we simply inspect the adjacency list. The gold standard, from a data structures perspective, would be near-linear time preprocessing and query answering in (poly)logarithmic time. Since this is unlikely for arbitrary $H$, we ask for well-motivated hypergraph properties that can make such a goal possible. We introduce bidegeneracy, a measure of hypergraph sparsity that strictly generalizes many other measures, including bounded degree, bounded rank, and bounded $\infty$-degeneracy. We show that bounded bidegeneracy yields fast hypergraph data structures. For a large set of fundamental queries, including their listing and sampling versions, we obtain near-linear preprocessing and (poly)logarithmic query time. Moreover, direct reductions from assumptions like the Orthogonal Vectors Conjecture suggest that, without assumptions on the input, these structures require almost-quadratic preprocessing or almost-linear query time. We use bidegeneracy to compute a compressed representation of $H$ and perform careful reductions to stabbing and searching problems in $\mathbb{R}^d$. Our framework significantly extends the well-studied "partial matching" problem of finding database entries through wildcard search. It also uncovers an intriguing connection between hypergraph sparsity, compression, and the complexity of hypergraph primitives. We also find remarkably low bidegeneracy in hypergraph datasets from the literature, suggesting practical consequences.

cs.DS↗

arXiv2026

Counting HyperGraphlets via Color Coding: a Quadratic Barrier and How to Break It

We study the problem of counting $k$-hypergraphlets, an interesting but surprisingly ignored primitive, with the aim of understanding whether efficient algorithms exist. To this end, we consider color coding, a well-known technique for approximately counting $k$-graphlets in graphs. Our first result is that, on hypergraphs, color coding encounters a quadratic barrier: under the Orthogonal Vector Conjecture, no implementation can run in sub-quadratic time in the input size. We then introduce a simple property, $(α,β)$-niceness, that hypergraphs from real-world datasets appear to satisfy for small values of $α$ and $β$. Intuitively, an $(α,β)$-nice hypergraph can be split into two sub-hypergraphs having respectively rank at most $α$ and degree at most $β$. By applying different techniques to each sub-hypergraph and carefully combining the outputs, we show how to run color coding in time $2^{O(k)} \cdot (2^β|V| + α^k |E| + α^2 β\|H\|)$, where $H=(V,E)$ is the input hypergraph. Afterwards, we can sample colorful $k$-hypergraphlets uniformly in expected $k^{O(k)} \cdot (β^2 + \ln |V|)$ time per sample. Experiments on real-world hypergraphs show that our algorithm significantly outperforms the naive quadratic algorithm, sometimes by more than an order of magnitude.

cs.DS↗