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 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.

Sketching Distances in Monotone Graph Classes

“…All proofs for adjacency sketching are in Section 3. Using standard random hashing and the adjacency labelling scheme of [KNR92], it is easy to see that any class of bounded arboricity is adjacency sketchable; this was stated explicitly in [Har20,HWZ21] (the latter giving slightly improved sketch size). Our contribution is to prove the converse for monotone classes (which does not hold for hereditary classes in general [HWZ21]).…”

“…Using standard random hashing and the adjacency labelling scheme of [KNR92], it is easy to see that any class of bounded arboricity is adjacency sketchable; this was stated explicitly in [Har20,HWZ21] (the latter giving slightly improved sketch size). Our contribution is to prove the converse for monotone classes (which does not hold for hereditary classes in general [HWZ21]). We use the probabilistic method to find a subgraph of small discrepancy in any class of unbounded arboricity, inspired by the recent proof of [HHH21a,HHH21b] that refuted the main conjecture of [HWZ21] (and we find that the simpler example of subgraphs of the hypercube also refutes the main conjecture of [HWZ21]).…”

“…A solution was suggested in [KNR92] and later conjectured in [Spi03], but recently refuted in a breakthrough of [HH21], leaving the problem wide open. Randomized adjacency labelling (i. e. adjacency sketching) was introduced in [FK09, Har20,HWZ21]. It was observed in [Har20,HWZ21] that a constant-size sketch implies an O(log n) labelling scheme, as desired in the above open problem.…”

“…Randomized adjacency labelling (i. e. adjacency sketching) was introduced in [FK09, Har20,HWZ21]. It was observed in [Har20,HWZ21] that a constant-size sketch implies an O(log n) labelling scheme, as desired in the above open problem. This raises the analogous problem of identifying the hereditary classes of graphs that admit constant-size adjacency sketches; it was further observed in [HWZ21] that this is equivalent to characterizing the Boolean-valued communication problems that admit constant-cost public-coin protocols, an open problem in communication complexity [HHH21b].…”

“…For r = 1, this coincides with adjacency labelling. The natural generalization of constant-size adjacency sketches is to ask for small-distance sketches whose size depends only on r; it was shown in [Har20] that such sketches exist for trees, and in [HWZ21] that they exist for any Cartesian product graphs and any stable 3 class of bounded twin-width (including, for example, planar graphs or any proper minor-closed class; see [GPT21]).…”