Tree-depth and the Formula Complexity of Subgraph Isomorphism

Kush, Deepanshu; Rossman, Benjamin

doi:10.1109/focs46700.2020.00012

Cited by 3 publications

(5 citation statements)

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“…Savitch [21] proved that s-t connectivity can be solved in O(log 2 (n)) space and n log 2 (n)(1+O(1)) time. Recent work of Kush and Rossman [16] has shown that the randomized AC 0 formula complexity of s-t connectivity is at most size n 0.49 log 2 (k)+O (1) , a slight improvement. Barnes, Buss, Ruzzo and Schieber [2] gave an algorithm running in both sublinear space and polynomial time for s-t connectivity.…”

Section: Related Work In Complexitymentioning

confidence: 99%

On the Number of Quantifiers as a Complexity Measure

Fagin¹,

Lenchner²,

Vyas³

et al. 2022

Preprint

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In 1981, Neil Immerman described a two-player game, which he called the "separability game" [14], that captures the number of quantifiers needed to describe a property in first-order logic. Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas [10] rediscovered the game, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer k, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank k, but where the same property needs 2 Ω(k 2 ) quantifiers to be expressed. We next give the precise number of quantifiers needed to distinguish two rooted trees of different depths. Finally, we give a new upper bound on the number of quantifiers needed to express s-t connectivity, improving the previous known bound by a constant factor.

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Section: Related Work In Complexitymentioning

confidence: 99%

On the Number of Quantifiers as a Complexity Measure

Fagin¹,

Lenchner²,

Vyas³

et al. 2022

Preprint

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show abstract

“…Remark 1. It is worth noting that in the Boolean setting, while there are known upper bounds and lower bounds based on these graph parameters for circuits and formulas for (colored) subgraph isomorphism problem, there is still an asymptotic gap between the upper bound and the corresponding lower bounds in these Boolean models (see [LRR17,KR20]). Moreover, our lower bound techniques are very different from those in the Boolean models.…”

Section: Our Contributionsmentioning

confidence: 99%

“…This, however, uses polynomial space in its execution. For a related problem, that is, detecting colored subgraph isomorphisms, such efficient space-time algorithms followed as a consequence of small-sized treedepth based Boolean formulas (see [KR20]).…”

Section: Our Contributionsmentioning

confidence: 99%

“…For instance, it is conjectured that the best known algorithms for variants of these pattern detection and counting problems can not be further improved. If true, this would imply P = NP in a rather strong way (see [LRR17,Mar10,Ros18,KR20]). Recently, it was discovered by Curticapean, Dell and Marx [CDM17] (also see [DST02]) that counting subgraph isomorphisms corresponds to counting linear combinations of homomorphisms, and hence establishing that it is sufficient to just consider the homomorphism counting problem.…”

Section: Introductionmentioning

confidence: 99%

“…These improved constructions for circuits and ABPs are one of the main source of advancement in finding improved algorithms for these pattern detection and counting problems [BKS18, KKM00,KLL13,FKLL15a,WWWY15]. The connection between the subgraph isomorphism problem and treedepth has also been explored in the context of Boolean computational models in [LRR17] and [KR20], where they discuss treewidth and treedepth based upper bounds and lower bounds for counting subgraph isomorphisms for Boolean circuits and formulas. In the context of parameterized counting complexity of these problems, a stronger connection between treewidth and the complexity of counting homomorphisms was established by Dalmau and Jonsson [DJ04] who showed a dichotomy theorem stating that when the pattern graph has bounded treewidth, then we can count homomorphisms in polynomial time, otherwise, we can show #W[1]-hardness.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Graph Homomorphism Polynomials: Algorithms and Complexity

Komarath¹,

Pandey²,

Rahul³

2020

Preprint

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We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP, VP, and VNP. We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively.Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. The characterization additionally yields new space-time efficient algorithms for several pattern detection and counting problems.

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