Implicit representations via the polynomial method

2026-02-11

AI Summary

This work addresses the design of compact adjacency labeling schemes for geometric graph classes such as semi-algebraic graphs, circumventing the exponential bit-cost incurred by high-precision coordinate representations. By introducing polynomial partitioning—a technique previously unused in this context—into label construction and leveraging the algebraic structure of adjacency predicates through tools from semi-algebraic set theory and combinatorial geometry, the authors achieve significant improvements in label length upper bounds. Specifically, they obtain labels of size $O(n^{1 - 2/(d+1) + \varepsilon})$ bits for $d$-dimensional semi-algebraic graphs, and $O(n^{1/3 + \varepsilon})$ bits for unit disk and segment intersection graphs. The framework further extends to new graph families, yielding $O(\log n)$-bit labels for semi-linear graphs and $O(\log^3 n)$-bit labels for polygon visibility graphs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon>0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

Problem

• semialgebraic graphs
• adjacency labeling schemes
• polynomial partitioning
• semilinear graphs
• polygon visibility graphs

Innovation

• polynomial partitioning
• adjacency labeling schemes
• semialgebraic graphs
• implicit representations
• semilinear graphs