Optimal and Efficient Partite Decompositions of Hypergraphs

TL;DR

This work settles tight asymptotics for partitioning graphs and $d$-uniform hypergraphs into complete partite substructures, achieving near-optimal per-vertex loads and total weights. It extends the Chung–Erdős–Spencer and Erdős–Pyber lines of results to hypergraphs, providing equitable decompositions with deterministic construction times and matching information-theoretic lower bounds. The authors also develop density-aware bounds, enabling succinct and compact biclique representations that support efficient independent-set, cut, and densest-subgraph queries, with direct implications for secret sharing and data compression. The results yield both theoretical optimality and practical algorithmic tools, and point to several directions for balanced hypergraph decompositions and dynamic/fractional extensions of the theory.

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 $γ\in (0,1)$ has a biclique partition of total weight at most $(\tfrac{1}{2}+o(1))\cdot h_2(γ) \frac{n^2}{\lg n}$, 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 $α> 1$.

Figures (5)

• Figure 1: Illustration of the application of biclique partitions in circuit design.
• Figure 2: Summary of our results. Dashed implications represent that other tools are also required. ($\blacksquare$ := density-aware, $\blacksquare$ := graph partitions, $\blacksquare$ := hypergraph partitions, $\blacksquare$ := finding bicliques)
• Figure 3: Illustration of the biclique partition construction.
• Figure 4: The 4 distributions for $d = 4$ up to permutation.
• Figure 5: Comparison of the leading constant for $\Omega(\lg n)$ balanced bicliques, and algorithmic runtimes.

Theorems & Definitions (73)

• theorem 1: Lupanov lupanov
• corollary 1
• theorem 2: chungDecompositionGraphsComplete1983
• theorem 3
• theorem 4
• theorem 5: Nechiporuk nechiporuk
• theorem 6
• theorem 7
• theorem 8
• definition 9