Randomized Communication and Implicit Graph Representations

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 subset of 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 "naively 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.