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 constantcost 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.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 satisfy the IGC, 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 Equality oracle, implying that the Equality 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 [Har20]. This generalizes to constant-size sketches for deciding firstorder formulas over the same graphs.
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.
We present an algorithm for testing halfspaces over arbitrary, unknown rotation-invariant distributions. Using O( √ nǫ −7 ) random examples of an unknown function f , the algorithm determines with high probability whether f is of the form f (x) = sign( i w i x i − t) or is ǫ-far from all such functions. This sample size is significantly smaller than the well-known requirement of Ω(n) samples for learning halfspaces, and known lower bounds imply that our sample size is optimal (in its dependence on n) up to logarithmic factors. The algorithm is distribution-free in the sense that it requires no knowledge of the distribution aside from the promise of rotation invariance. To prove the correctness of this algorithm we present a theorem relating the distance between a function and a halfspace to the distance between their centers of mass, that applies to arbitrary distributions. *
We study the problems of adjacency sketching, small-distance sketching, and approximate distance threshold sketching for monotone classes of graphs. The problem is to obtain randomized sketches of the vertices of any graph G in the class, so that adjacency, exact distance thresholds, or approximate distance thresholds of two vertices u, v can be decided (with high probability) from the sketches of u and v, by a decoder that does not know the graph. The goal is to determine when sketches of constant size exist.We show that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false. The existence of an adjacency sketch is equivalent to the condition of bounded arboricity, while the existence of small-distance sketches is equivalent to the condition of bounded expansion. Classes of constant expansion admit approximate distance threshold sketches, while a monotone graph class can have arbitrarily small non-constant expansion without admitting an approximate distance threshold sketch. * Partially supported by the French ANR Projects GATO (ANR-16-CE40-0009-01), GrR (ANR-18-CE40-0032), TWIN-WIDTH (ANR-21-CE48-0014-01) and by LabEx PERSYVAL-lab (ANR-11-LABX-0025).† This work was partly funded by an NSERC Canada Graduate Scholarship.
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