Twin-Width is Linear in the Poset Width

Balabán, Jakub; Hliněný, Petr

doi:10.48550/arxiv.2106.15337

Cited by 2 publications

(4 citation statements)

References 8 publications

(20 reference statements)

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“…Often, graph families where a certain width parameter is bounded possess favorable structural, algorithmic, or combinatorial properties. The powerful twin-width parameter, introduced recently in [15], generalizes treewidth and clique-width and has attracted a lot of recent attention [1,8,25,50,51]. Graph families of bounded twin-width are positive examples to the IGQ [14], making them a natural choice for studying Question 1.2.…”

Section: Resultsmentioning

confidence: 99%

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: Resultsmentioning

confidence: 99%

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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“…Secondly, suppose that t ă s ă t `r. By the construction of G s , if v " v i,1 for some i P rk 1 `?q ln q `?q `2 ln q 2 `1 " 3 2…”

Section: Twin-width Versus the Number Of Edgesmentioning

confidence: 99%

“…It has been shown that every graph class of bounded twin-width is small 3 [4], χ-bounded [5], and admits a linear-time algorithm for first-order model checking [8] (if we are given a certificate for an input graph to have small twin-width). Despite being a relatively new concept, twin-width has already generated a large amount of interests [8,4,5,6,13,7,1,10,17,9].…”

Section: Introductionmentioning

confidence: 99%

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Bounds for the Twin-width of Graphs

Ahn,

Hendrey,

Kim

et al. 2021

Preprint

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Bonnet, Kim, Thomassé, and Watrigant [8] introduced the twinwidth of a graph. We show that the twin-width of an n-vertex graph is less than pn `?n ln n `?n `2 ln nq{2, and the twin-width of an m-edge graph is less than ? 3m `m1{4 ? ln m{p4 ¨31{4 q `3m 1{4 {2. Conference graphs of order n (when such graphs exist) have twinwidth at least pn ´1q{2, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph Gpn, pq with 1{n ď p " ppnq ď 1{2 is larger than 2pp1 ´pqn ´p2 ? 2 `εq a pp1 ´pqn ln n asymptotically almost surely for any positive ε. Lastly, we calculate the twin-width of random graphs Gpn, pq with p ď c{n for a constant c ă 1, determining the thresholds at which the twin-width jumps from 0 to 1 and from 1 to 2.

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Twin-Width is Linear in the Poset Width

Cited by 2 publications

References 8 publications

Randomized communication and implicit graph representations

Randomized communication and implicit graph representations

Bounds for the Twin-width of Graphs

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