On the $AC^0$ Complexity of Subgraph Isomorphism

Li, Yuan; Razborov, Alexander A.; Rossman, Benjamin

doi:10.1137/14099721x

Cited by 22 publications

(49 citation statements)

References 14 publications

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“…Amano [3] generalized the technique to the G-subgraph isomorphism problem for arbitrary patterns G and also gave an extension to hypergraphs. Subsequent work of Li, Razborov and the author [24] further generalized the technique to a colored variant of the G-subgraph isomorphism problem, obtaining an n Ω(w/ log w) lower bound for patterns of tree-width w. This result is presented in §4.…”

Section: Outline Of the Articlementioning

confidence: 99%

Lower Bounds for Subgraph Isomorphism

Rossman¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We consider the problem of determining whether an Erdős-Rényi random graph contains a subgraph isomorphic to a fixed pattern, such as a clique or cycle of constant size. The computational complexity of this problem is tied to fundamental open questions including P vs. NP and NC 1 vs. L. We give an overview of unconditional averagecase lower bounds for this problem (and its colored variant) in a few important restricted classes of Boolean circuits.

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Section: Outline Of the Articlementioning

confidence: 99%

Lower Bounds for Subgraph Isomorphism

Rossman¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and logic. By combining our polynomial excluded-minor approximation of treedepth with lower bounds on the AC 0 formula size of detecting grids [18], paths [27] and trees [29], we obtain an n poly(td(G)) lower bound on the AC 0 formula size of the colored G-subgraph isomorphism problem for all graphs G. This result, in turn, has a surprising corollary in finite model theory: a polynomial-rank homomorphism preservation theorem on finite structures. In this section, we give a brief overview of these results; see the paper [28] for details.…”

Section: Applications In Complexity and Logicmentioning

confidence: 99%

“…The following lemma from Li et al [18] shows that the complexity of SUB(G) is a minor-monotone graph invariant.…”

Section: The Acmentioning

confidence: 99%

A Polynomial Excluded-Minor Approximation of Treedepth

Kawarabayashi

Rossman

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Treedepth is a well-studied graph invariant in the family of "width measures" that includes treewidth and pathwidth. Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k × k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths. Specifically, we show that there is a constant c such that every graph of treedepth ≥ k c contains one of the following minors (each of treedepth ≥ k):• the complete binary tree of height k,• the path of order 2 k .Let us point out that we cannot drop any of the above graphs for our purpose. Moreover, given a graph G we can, in randomized polynomial time, find either an embedding of one of these minors or conclude that treedepth of G is at most k c .This result has potential applications in a variety of settings where bounded treedepth plays a role. In addition to some graph structural applications, we describe a surprising application in circuit complexity and finite model theory from recent work of the second author [28].

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“…The maximum ρ(K) over all subgraphs K of a graph F will be denoted by ρ * (F ). The following fact from the random graph theory was used also in [20] for proving average-case lower bounds on the AC 0 complexity of Subgraph Isomorphism. With high probability means the probability approaching 1 as n → ∞.…”

Section: Definitions Over Highly Connected Graphsmentioning

confidence: 99%

The Descriptive Complexity of Subgraph Isomorphism Without Numerics

Verbitsky

Zhukovskii

2018

Theory Comput Syst

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Let F be a connected graph with ℓ vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than ℓ, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs K ℓ and K ℓ−1 . We show that, for some F , the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth ℓ − 3. On the other hand, this is never possible with quantifier depth better than ℓ/2. If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some F be arbitrarily small comparing to ℓ but never smaller than the treewidth of F . Moreover, the definitions over highly connected graphs require quantifier depth strictly more than the density of F . Finally, we determine the exact values of these descriptive complexity parameters for all connected pattern graphs F on 4 vertices.

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On the $AC^0$ Complexity of Subgraph Isomorphism

Cited by 22 publications

References 14 publications

Lower Bounds for Subgraph Isomorphism

Lower Bounds for Subgraph Isomorphism

A Polynomial Excluded-Minor Approximation of Treedepth

The Descriptive Complexity of Subgraph Isomorphism Without Numerics

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