Implicit representations via the polynomial method

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 $O(\log^3 n)$.

💡 Research Summary

The paper addresses the problem of compact implicit representations for families of semialgebraic graphs—graphs whose vertices are points in ℝ^d and whose edges are defined by a Boolean combination of a constant‑size set of polynomial sign conditions. While such graphs naturally admit a geometric representation, converting this into a binary labeling scheme is non‑trivial because some realizations require doubly‑exponential coordinate precision. The authors develop a new technique based on modern polynomial partitioning (the Guth‑Katz style) to construct balanced biclique decompositions of any d‑dimensional semialgebraic graph.

A biclique decomposition partitions the edge set into complete bipartite subgraphs. By assigning each vertex the list of bicliques it belongs to (together with a one‑bit side indicator), an adjacency test reduces to intersecting two short lists. The total label size is proportional to ν(n)·O(log n), where ν(n) is the maximum number of bicliques incident to any vertex. The main technical contribution is to bound ν(n) by O(n^{1‑2/(d+1)+ε}) for any ε>0, using recursive polynomial partitioning that splits the point set into O(r) cells, each containing roughly n/r points, and recurses inside each cell. This yields a biclique decomposition of total size O(n^{2d/(d+1)+ε}) and a perfectly balanced distribution of bicliques among vertices.

From this, the authors obtain their central theorem: every d‑dimensional semialgebraic family admits an adjacency labeling scheme using O(n^{1‑2/(d+1)+ε}) bits per vertex. Several concrete corollaries follow. For unit‑disk graphs (intersection graphs of unit disks in the plane) and segment‑intersection graphs, the effective “parameter dimension” relevant to the intersection predicate is 2, not the ambient 4, leading to O(n^{1/3+ε})‑bit labels. Disk graphs with arbitrary radii receive O(n^{1/2+ε})‑bit labels, and more generally unit‑ball graphs in ℝ^d get O(n^{1‑2/(d+1)+ε})‑bit labels, while arbitrary ball graphs obtain O(n^{1‑2/(d+2)+ε})‑bit labels.

For semilinear graphs—semialgebraic graphs whose defining polynomials are linear—the authors show a dramatically stronger result: an O(log n)‑bit labeling scheme suffices. The proof decomposes a semilinear graph into a constant number of comparability graphs of bounded dimension; each comparability graph can be encoded by the rank of each vertex in each dimension, yielding logarithmic labels. This covers many well‑studied families such as interval graphs, permutation graphs, cographs, bounded‑boxicity graphs, circle graphs, and distance‑hereditary graphs.

Polygon visibility graphs are not semialgebraic in the strict sense because adjacency depends on the entire polygon. Nevertheless, using a result of Agarwal, Alon, Aronov, and Suri on 3‑dimensional range searching, the authors construct a labeling scheme with O(log³ n) bits per vertex. A refined analysis gives O(log² n) bits for capped graphs, which include terrain visibility graphs.

The paper also discusses the limitations of natural geometric representations: some semialgebraic families (e.g., point‑line incidence graphs derived from the Perles configuration) require irrational or doubly‑exponential coordinates, making direct encoding infeasible. The polynomial‑partition based labeling circumvents this issue, providing a purely combinatorial encoding.

Finally, the authors note that the biclique decomposition size they achieve matches the theoretical lower bound implied by the size of any labeling derived from such a decomposition, namely Ω(n^{1‑2/(d+1)}). They leave open the problem of proving stronger lower bounds for semialgebraic families and of tightening the relationship between graph speed, dimension, and labeling size. Overall, the work significantly advances the state of the art in implicit graph representations for geometric and algebraic graph families.