Optimal and Efficient Partite Decompositions of Hypergraphs

2025-11-14

AI Summary

This paper studies edge-set decomposition of $d$-uniform hypergraphs: partitioning the edge set into a family $mathcal{F}$ of complete $d$-partite $d$-uniform hypergraphs (i.e., $d$-cliques), minimizing the maximum vertex participation—i.e., the largest number of hyperedges in $mathcal{F}$ containing any single vertex. This problem generalizes the Erdős–Pyber conjecture on graph decomposition and resolves a 40-year-old open question posed by Chung, Erdős, and Spencer (1983). The authors establish an asymptotically optimal upper bound $(1/d! + o_d(1)) n^{d-1} / log n$ on vertex participation. They introduce a constructive method based on binary entropy $h_2(gamma)$, yielding an $O(m)$-time biclique decomposition; and design an $n^{1+o(1)}$-time algorithm to efficiently find large balanced complete subhypergraphs. These results yield information-theoretically optimal binary secret sharing schemes, enable $O(n^2/log n)$-time responses to independent set and cut queries, and facilitate efficient approximation of the densest subhypergraph.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the problem of partitioning the edges of a $d$-uniform hypergraph $H$ into a family $F$ of complete $d$-partite hypergraphs ($d$-cliques). We show that there is a partition $F$ in which every vertex $v in V(H)$ belongs to at most $(frac{1}{d!} + o_d(1))n^{d-1}/lg n$ members of $F$. This settles the central question of a line of research initiated by Erdős and Pyber (1997) for graphs, and more recently by Csirmaz, Ligeti, and Tardos (2014) for hypergraphs. The $d=2$ case of this theorem answers a 40-year-old question of Chung, Erdős, and Spencer (1983). An immediate corollary of our result is an improved upper bound for the maximum share size for binary secret sharing schemes on uniform hypergraphs. Building on results of Nechiporuk (1969), we prove that every graph with fixed edge density $γin (0,1)$ has a biclique partition of total weight at most $( frac{1}{2}+o(1))cdot h_2(γ) frac{n^2}{lg n}$, where $h_2$ is the binary entropy function. Our construction implies that such biclique partitions can be constructed in time $O(m)$, which answers a question of Feder and Motwani (1995) and also improves upon results of Mubayi and Turán (2010) as well as Chavan, Rabinia, Grosu, and Brocanelli (2025). Using similar techniques, we also give an $n^{1+o(1)}$ algorithm for finding a subgraph $K_{t,t}$ with $t = (1-o(1)) fracγ{h_2(γ)} lg n$. Our results show that biclique partitions are information-theoretically optimal representations for graphs at every fixed density. We show that with this succinct representation one can answer independent set queries and cut queries in time $O(n^2/ lg n)$, and if we increase the space usage by a constant factor, we can compute a $2α$-approximation for the densest subgraph problem in time $O(n^2/lg α)$ for any $α> 1$.

Problem

• Partition hypergraph edges into complete d-partite hypergraphs efficiently
• Determine optimal biclique partitions for graphs with fixed edge density
• Develop efficient algorithms for independent set and cut queries

Innovation

• Partition hypergraph edges into complete d-partite hypergraphs
• Construct biclique partitions with total weight O(n²/lg n)
• Enable efficient graph queries using succinct biclique representations