Randomized communication and implicit graph representations

Abstract

The most basic lower-bound question in randomized communication complexity is: Does a given problem have constant cost, or non-constant cost? We observe that this question has a deep connection to implicit graph representations in structural graph theory. Specifically, constant-cost communication problems correspond to hereditary graph families that admit constant-size adjacency sketches, or equivalently constant-size probabilistic universal graphs (PUGs), and these graph families are a subset of families that admit adjacency labeling schemes of size 𝑂 (log 𝑛), which are the subject of the well-studied implicit graph question (IGQ).We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) giving a structural characterization of randomized constant-cost communication problems, and (2) resolving a probabilistic version of the IGQ. For each family F studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies H ⊆ F are either stable (in a sense relating to model theory), in which case they admit constant-size PUGs (i.e. adjacency sketches), or they are not stable, in which case they do not.The correspondence between communication problems and hereditary graph families allows for a probabilistic method of constructing adjacency labeling schemes. By this method, we show that the induced subgraphs of any Cartesian products 𝐺 𝑑 are positive examples to the IGQ, also giving a bound on the number of unique induced subgraphs of any graph product. We prove that this probabilistic construction cannot be "naïvely derandomized" by using an Eqality oracle, implying that the Eqality oracle cannot simulate the 𝑘-Hamming Distance communication protocol.As a consequence of our results, we obtain constant-size sketches for deciding dist(𝑥, 𝑦) ≤ 𝑘 for vertices 𝑥, 𝑦 in any stable graph family with bounded twin-width, answering an open question about planar graphs from earlier work. This generalizes to constant-size sketches for deciding first-order formulas over the same graphs.

Graph parameters, implicit representations and factorial properties

“…This extends to a transformation between hereditary classes: transform every graph in a hereditary class to a bipartite graph and take the hereditary closure of the obtained set of bipartite graphs. As was shown in [18], this transformation preserves the factorial speed of growth as well as the existence of an implicit representation 1 .…”

“…Indeed, in [2] it was shown that Q has unbounded functionality. On the other hand, it was shown in [17] that the class admits an implicit representation and is, in particular, factorial; in fact, more generally, the hereditary closure of Cartesian products of any finite set of graphs [18] and even of any class admitting an implicit representation [14], admits an implicit representation. These results, however, are non-constructive and they provide neither explicit labeling schemes, nor specific factorial bounds on the number of graphs.…”

Optimal Adjacency Labels for Subgraphs of Cartesian Products

“…This phase is an easy extension of the simple proof in [Har22] of the labeling scheme for induced subgraphs of hypercubes, derived from [Har20,HWZ22]. The labels are obtained by a simple probabilistic method (and are efficiently computable by a randomized algorithm).…”

“…The XOR-labeling trick can also be used to simplify the proof of [HWZ22] for adjacency sketches of Cartesian products. That proof uses a two-level hashing scheme to avoid destroying the labels of x i and y i with the XOR, with a high constant probability of success.…”

“…Meanwhile, the subgraphs of hypercubes are a counterexample [EHK22] to a conjecture in [HWZ22] about adjacency sketches (a randomized variant of labeling schemes). Cartesian products are important for understanding the relationship between sketches and labels: a constant-size sketch and an O(log n) labeling scheme exist for induced subgraphs of the hypercube, but constantsize sketches for subgraphs do not exist [EHK22].…”