On Zarankiewicz's Problem for Intersection Hypergraphs of Geometric Objects

2024-12-09

International Symposium on Computational Geometry

AI Summary

This work addresses the Zarankiewicz problem for geometric $r$-partite intersection hypergraphs: given a family of axis-parallel boxes or pseudo-disks in $mathbb{R}^d$, determine the maximum number of hyperedges in its $r$-partite intersection hypergraph that avoids a complete $r$-partite subhypergraph $K_{t,dots,t}$. Methodologically, we integrate refined graph-theoretic Zarankiewicz bounds with shallow-cutting, bipartite subgraph covering, transversality, and planarity arguments—bypassing reliance on algebraic structure. Our main contributions are: (1) For $d$-dimensional axis-parallel boxes, we establish the first tight-order upper bound $Oig(n^{r-1} (log n / log log n)^{d-1}ig)$, improving prior results by a factor of approximately $(log n)^{d(2^{r-1}-2)}$; (2) For pseudo-disk families, we obtain $Oig(n^{r-1} (log n)^{r-2}ig)$, substantially outperforming the best general bound from the past six decades and semi-algebraic approaches, with an improvement of order $n^{ ilde{Omega}((2r-2)/(3r-2))}$.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The hypergraph Zarankiewicz's problem, introduced by ErdH{o}s in 1964, asks for the maximum number of hyperedges in an $r$-partite hypergraph with $n$ vertices in each part that does not contain a copy of $K_{t,t,ldots,t}$. ErdH{o}s obtained a near optimal bound of $O(n^{r-1/t^{r-1}})$ for general hypergraphs. In recent years, several works obtained improved bounds under various algebraic assumptions -- e.g., if the hypergraph is semialgebraic. In this paper we study the problem in a geometric setting -- for $r$-partite intersection hypergraphs of families of geometric objects. Our main results are essentially sharp bounds for families of axis-parallel boxes in $mathbb{R}^d$ and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound $O_{d,t}(n^{r-1}(frac{log n}{log log n})^{d-1})$. The best previous bound was larger by a factor of about $(log n)^{d(2^{r-1}-2)}$. For pseudo-discs, we obtain the bound $O_t(n^{r-1}(log n)^{r-2})$, which is sharp up to logarithmic factors. As this hypergraph has no algebraic structure, no improvement of ErdH{o}s' 60-year-old $O(n^{r-1/t^{r-1}})$ bound was known for this setting. Futhermore, even in the special case of discs for which the semialgebraic structure can be used, our result improves the best known result by a factor of $ ilde{Omega}(n^{frac{2r-2}{3r-2}})$. To obtain our results, we use the recently improved results for the graph Zarankiewicz's problem in the corresponding settings, along with a variety of combinatorial and geometric techniques, including shallow cuttings, biclique covers, transversals, and planarity.

Problem

• Studying hypergraph Zarankiewicz problem for geometric intersection hypergraphs
• Obtaining sharp bounds for axis-parallel boxes and pseudo-discs families
• Improving previous bounds using combinatorial and geometric techniques

Innovation

• Axis-parallel boxes with sharp logarithmic bounds
• Pseudo-discs with near-sharp logarithmic factors
• Combinatorial and geometric techniques including shallow cuttings