Induced subgraph isomorphism: Are some patterns substantially easier than others?

Floderus, Peter; Kowaluk, Mirosław; Lingas, Andrzej; Lundell, Eva-Marta

doi:10.1016/j.tcs.2015.09.001

Cited by 24 publications

(16 citation statements)

References 15 publications

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“…One of the few results related to this is by Floderus et al [16] who showed that if a pattern H contains a t-Clique that is disjoint from all other t-Cliques in H, then H is at least as hard to detect as a t-Clique. This implied strong clique-based hardness results for induced k-path and k-cycle.…”

Section: Hardnessmentioning

confidence: 99%

“…This implied strong clique-based hardness results for induced k-path and k-cycle. However, the reduction of [16] fails for patterns whose k-Cliques intersect non-trivially.…”

Section: Hardnessmentioning

confidence: 99%

“…The main difficulty in reducing k-Clique to the detection problem for other graph patterns H can be seen in the following natural attempt used e.g. by [16]. Say H has a k-clique K and let H be the graph induced by the vertices of H not in K. Let G = (V, E) be an instance of k-Clique.…”

Section: Hardnessmentioning

confidence: 99%

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Graph pattern detection: hardness for all induced patterns and faster non-induced cycles

Dalirrooyfard

Vuong

Williams

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We consider the pattern detection problem in graphs: given a constant size pattern graph H and a host graph G, determine whether G contains a subgraph isomorphic to H. We present the following new improved upper and lower bounds:• We prove that if a pattern H contains a k-clique subgraph, then detecting whether an n node host graph contains a not necessarily induced copy of H requires at least the time for detecting whether an n node graph contains a k-clique. The previous result of this nature required that H contains a k-clique which is disjoint from all other k-cliques of H.• We show that if the famous Hadwiger conjecture from graph theory is true, then detecting whether an n node host graph contains a not necessarily induced copy of a pattern with chromatic number t requires at least the time for detecting whether an n node graph contains a t-clique. This implies that: (1) under Hadwiger's conjecture for every k-node pattern H, finding an induced copy of H requires at least the time of √ k-clique detection and size ω(n √ k/4 ) for any constant depth circuit, and (2) unconditionally, detecting an induced copy of a random G(k, p) pattern w.h.p. requires at least the time of Θ(k/ log k)-clique detection, and hence also at least size n Ω(k/ log k) for circuits of constant depth.• We show that for every k, there exists a k-node pattern that contains a k − 1-clique and that can be detected as an induced subgraph in n node graphs in the best known running time for k − 1-Clique detection. Previously such a result was only known for infinitely many k.• Finally, we consider the case when the pattern is a directed cycle on k nodes, and we would like to detect whether a directed m-edge graph G contains a k-Cycle as a not necessarily induced subgraph. We resolve a 14 year old conjecture of [Yuster-Zwick SODA'04] on the complexity of k-Cycle detection by giving a tight analysis of their k-Cycle algorithm. Our analysis improves the best bounds for k-Cycle detection in directed graphs, for all k > 5.

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Section: Hardnessmentioning

confidence: 99%

“…This implied strong clique-based hardness results for induced k-path and k-cycle. However, the reduction of [16] fails for patterns whose k-Cliques intersect non-trivially.…”

Section: Hardnessmentioning

confidence: 99%

See 1 more Smart Citation

Graph pattern detection: hardness for all induced patterns and faster non-induced cycles

Dalirrooyfard

Vuong

Williams

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…claw, diamond and their complements) one can give an efficient reduction from triangle detection (see e.g. Floderus et al [FKLL12]). These reductions imply that our algorithms are optimal unless there is a breakthrough on triangle detection.…”

Section: Introductionmentioning

confidence: 99%

Finding Four-Node Subgraphs in Triangle Time

Williams¹,

Wang²,

Williams³

et al. 2014

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

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We present new algorithms for finding induced four-node subgraphs in a given graph, which run in time roughly that of detecting a clique on three nodes (i.e., a triangle).• The best known algorithms for triangle finding in an nnode graph take O(n ω ) time, where ω < 2.373 is the matrix multiplication exponent. We give a general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, iñ O(n ω ) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministicÕ(n ω ) time.Our approach substantially improves on prior work. For instance, the previous best algorithm for C 4 detection ran in O(n 3.3 ) time, and for diamond detection in O(n 3 ) time.• For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m 2ω/(ω+1) ) ≤ O(m 1.41 ) time. We give a randomizedÕ(m 2ω/(ω+1) ) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C 4 , K 4 and their complements. In the case of diamond detection, we also design a deterministicÕ(m 2ω/(ω+1) ) time algorithm. For C 4 or its complement, we give randomized O(m (4ω−1)/(2ω+1) ) ≤ O(m 1.48 ) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m 1.5 ) time.

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“…The state-of-the-art of the algorithmics for Induced Subgraph Isomorphism is different from Subgraph Isomorphism. Floderus et al [13] collected evidences in favour of the conjecture that I(F ) for F with ℓ vertices cannot be recognized faster than I(K c ℓ ), where c < 1 is a constant. Similarly to D(F ), we use notation D[F ] = D(I(F )) and W [F ] = W (I(F )), where the square brackets indicate that the case of induced subgraphs is considered.…”

Section: Introductionmentioning

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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Induced subgraph isomorphism: Are some patterns substantially easier than others?

Cited by 24 publications

References 15 publications

Graph pattern detection: hardness for all induced patterns and faster non-induced cycles

Graph pattern detection: hardness for all induced patterns and faster non-induced cycles

Finding Four-Node Subgraphs in Triangle Time

The Descriptive Complexity of Subgraph Isomorphism Without Numerics

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