Randomized Communication and the Implicit Graph Conjecture
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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 the Implicit Graph Conjecture (IGC) in structural graph theory. Specifically, constant-cost communication problems correspond to a certain subset of hereditary graph families that satisfy the IGC: those that admit constant-size probabilistic universal graphs (PUGs), or, equivalently, those that admit constant-size adjacency sketches. 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 IGC. 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. We conjecture that this always holds, i.e. that constant-cost randomized communication problems correspond to the set of stable families that satisfy the IGC. As a consequence of our results, we also obtain constant-size adjacency sketches, and an O(log n) adjacency labeling scheme, for the induced subgraphs of arbitrary Cartesian products, as well as constant-size small-distance sketches for Cartesian products and stable families of bounded twin-width (including planar graphs, answering a question from earlier work).
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