Sketching Distances in Monotone Graph Classes

Harms, Nathaniel; Kupavskii, Andrey

doi:10.48550/arxiv.2202.09253

Cited by 1 publication

(2 citation statements)

References 35 publications

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“…There are now two counterexamples to our PUG conjecture, in [23,27]. Soon after we released a preprint of the current paper, Hambardzumyan, Hatami, & Hatami [27] refuted our conjecture along with its unintuitive consequence, using an interesting probabilistic construction of a hereditary graph family derived from their (independent and concurrent) work on constant-cost communication complexity [28].…”

Section: Discussion and Subsequentmentioning

confidence: 99%

“…This leaves wide open both the IGQ and Question 1.2, which we argue should be studied in parallel, due to the connections presented here. Question 1.2 may even be a more approachable step towards the IGQ, because communication complexity can give lower bounds on sketches [23,27], whereas for labeling schemes there are not yet any lower bound techniques against fixed hereditary families of factorial speed. Another question raised by our results is, why does stability characterize the existence of constant-size PUGs, among so many important families in the literature on the IGQ?…”

Section: Discussion and Subsequentmentioning

confidence: 99%

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Randomized communication and implicit graph representations

Harms

Wild

Zamaraev

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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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 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.

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Section: Discussion and Subsequentmentioning

confidence: 99%

Section: Discussion and Subsequentmentioning

confidence: 99%