Randomized Communication and Implicit Graph Representations

TL;DR

This work investigates the landscape of constant-cost randomized communication problems, establishing a rigorous equivalence with hereditary graph classes that admit constant-size adjacency sketches and probabilistic universal graphs (PUGs). It develops two complementary viewpoints: a communication-theoretic one and a graph-representation perspective, revealing that constant-cost problems correspond to stable, factorial graph classes whose graphs admit constant-size PUGs; it also shows that Cartesian products preserve constant-size sketches, enabling new implicit representations for product graphs. The paper provides several non-trivial constant-cost examples (e.g., small-distance predicates in planar graphs and distance-1 problems in Cartesian products) and identifies stability as a key structural criterion explaining the existence of constant-cost protocols. It also proves Equality is not complete for this class by showing 1-Hamming Distance does not reduce to Equality, clarifying the boundary between constant-cost problems and those reducible to Equality. The results advance both the understanding of implicit graph representations and the design of efficient randomized communication protocols, with implications for distance sketches, graph-product constructions, and the broader implicit-graph conjecture landscape.

Abstract

We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed under taking induced subgraphs) graph classes which admit constant-size adjacency sketches and probabilistic universal graphs (PUGs), which are randomized versions of the well-studied adjacency labeling schemes and induced-universal graphs. This gives a new perspective on long-standing questions about the existence of these objects, including new methods of constructing adjacency labeling schemes. We ask three main questions about constant-cost communication, or equivalently, constant-size PUGs: (1) Are there any natural, non-trivial problems aside from Equality and k-Hamming Distance which have constant-cost communication? We provide a number of new examples, including deciding whether two vertices have path-distance at most k in a planar graph, and showing that constant-size PUGs are preserved by the Cartesian product operation. (2) What structures of a problem explain the existence or non-existence of a constant-cost protocol? We show that in many cases a Greater-Than subproblem is such a structure. (3) Is the Equality problem complete for constant-cost randomized communication? We show that it is not: there are constant-cost problems which do not reduce to Equality.

Figures (9)

• Figure 4: The correspondence that motivates this paper (\ref{['prop:correspondence 1']}). \ref{['section:graph theory']} describes the lattice on the right. Communication problems with constant-cost randomized protocols are mapped to the set of hereditary graph classes with constant-size PUGs (and therefore $\mathop{\mathrm{poly}}\nolimits(n)$ universal graphs by \ref{['prop:correspondence inclusion']}) by $\mathfrak{F}$. Families with constant-size PUGs are mapped to constant-cost communication problems by $\textsc{Adj}$.
• Figure 5: Examples of the half-graph, co-half-graph, and threshold graphs.
• Figure 6: Small-distance sketch for Cartesian products. Along the bottom are the distances sketches for $x_i$ (top) and $y_i$ (bottom) for $i=1$ to $d$. Where $x_i=y_i$, the sketches for $x_i$ and $y_i$ are equal and are colored grey; they cancel out in $Z$ and $z$. Where $x_i \neq y_i$, the sketch for $x_i$ is in red and the sketch for $y_i$ is in green. For $i \neq j$ where $x_i \neq y_i$ and $x_j \neq y_j$, the sketches for $x_i,y_i$ and $x_j,y_j$ are mapped to different buckets with high probability (i. e., $b(i) \neq b(j)$) and the sketches for $x_i$ and $y_i$ are mapped to different locations in the same bucket with high probability (i. e., $c(i,x_i) \neq c(i,y_i)$).
• Figure 8: The permutation graph decomposition.
• Figure 9: The bipartite graphs from \ref{['def:S123-P7-Fpq']}

Theorems & Definitions (131)

• Theorem 1.1: Informal; see \ref{['section:twin-width']}
• Theorem 1.2: \ref{['section:cartesian product']}
• Example 1.3
• Theorem 1.4: Informal; \ref{['section:intersection graphs']}
• Theorem 1.5: Informal; \ref{['section:bipartite graphs']}
• Theorem 1.6: \ref{['section:equality']}
• definition 2.1
• definition 2.2: Adjacency labeling scheme
• definition 2.3
• proposition 2.4: KNR92