Tight Bounds for Graph Homomorphism and Subgraph Isomorphism

Cygan, Marek; Fomin, Fedor V.; Golovnev, Alexander; Mihajlin, Ivan; Pachocki, Jakub; Socała, Arkadiusz

doi:10.1137/1.9781611974331.ch112

Cited by 13 publications

(10 citation statements)

References 9 publications

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“…Corollary 2 in particular excludes any algorithm for testing homomorphisms into Kneser graphs with running time 2 O(n+h) . It cannot give a tight lower bound matching the result of Cygan et al [9] for general homomorphisms, because h = |V (KG a,b )| = a b is not polynomial in b. On the other hand, it exhibits the first explicit family of graphs H for which the complexity of Graph Homomorphism increases with h.…”

Section: Corollarymentioning

confidence: 82%

“…For quite a while it was open whether there is an algorithm for Graph Homomorphism running in time 2 O(n+h) . It was recently answered in the negative by Cygan et al [9]; more precisely, they proved that an algorithm with running time 2 o(n log h) would contradict the Exponential Time Hypothesis (ETH) of Impagliazzo et al [23]. However, Graph Homomorphism is a very general problem, hence researchers try to uncover a more fine-grained picture and identify families of graphs H such that the problem can be solved more efficiently whenever H ∈ H. For example, Fomin, Heggernes and Kratsch [13] showed that when H is of treewidth at most t, then Graph Homomorphism can be solved in time O ⋆ ((t + 3) n ).…”

Section: Introductionmentioning

confidence: 99%

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Tight Lower Bounds for the Complexity of Multicoloring

Bonamy

Kowalik

Pilipczuk

et al. 2019

ACM Trans. Comput. Theory

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In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b = 1 case) is equivalent to finding a homomorphism to the Kneser graph KG a,b , and gives relaxations approaching the fractional chromatic number.We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with running time f (b)·2 o(log b)·n , for any computable f (b), unless the Exponential Time Hypothesis (ETH) fails. A (b + 1) n · poly(n)-time algorithm due to Nederlof [2008] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2 O(n+h) algorithm unless ETH fails, even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [SODA 2016].The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH.1 The O ⋆ (·) notation hides factors polynomial in the input size.

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Section: Corollarymentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Tight Lower Bounds for the Complexity of Multicoloring

Bonamy

Kowalik

Pilipczuk

et al. 2019

ACM Trans. Comput. Theory

Self Cite

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“…By Theorem 13 it then follows that for every |D| ≥ 3 there exists B such that CSP(R D = )-B is NP-complete but not solvable in subexponential time, unless the ETH is false. The ETH-hardness for degree-bounded k-coloring is typically referred to as folklore (see, e.g., Cygan et al [22,Lemma 1]), but our results show that this property can be phrased in a purely algebraic manner.…”

Section: Complexity Of Csps In Light Of the Ethmentioning

confidence: 60%

Fine-Grained Time Complexity of Constraint Satisfaction Problems

Jönsson

Lagerkvist

Roy³

2021

ACM Trans. Comput. Theory

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We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

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“…The idea of using harmonious coloring for proving lower bounds of the form 2 o( 2 ) n O (1) was used by Agrawal et al [1] to prove a lower bound for Split Contraction, when parameterized by the vertex cover number , of the input graph. Also, the idea of partitioning vertices 60:3 of the input graph based on some coloring scheme was used by Cygan et al [10] to prove ETH-based lower bounds for Graph Homomorphism and Subgraph Isomorphism.…”

Section: Inputmentioning

confidence: 99%

Fine-grained complexity of rainbow coloring and its variants

Agrawal

2022

Journal of Computer and System Sciences

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Consider a graph G and an edge-coloring c R : E(G) → [k]. A rainbow path between u, v ∈ V (G) is a path P from u to v such that for all e, e ∈ E(P ), where e = e we have c R (e) = c R (e ). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c R :there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S ⊆ V (G) × V (G), and the objective is to decide if there exists c R : E(G) → [k] such that for all (u, v) ∈ S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S ⊆ V (G), and the objective is to decide if there exists c R : E(G) → [k] such that for all u, v ∈ S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results. For every k ≥ 3, Rainbow k-Coloring does not admit an algorithm running in time 2 o(|E(G)|) n O(1) , unless ETH fails. For every k ≥ 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2 o(|S| 2 ) n O(1) , unless ETH fails. Subset Rainbow k-Coloring admits an algorithm running in time 2 O(|S|) n O (1) . This also implies an algorithm running in time 2 o(|S| 2 ) n O(1) for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.

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Tight Bounds for Graph Homomorphism and Subgraph Isomorphism

Cited by 13 publications

References 9 publications

Tight Lower Bounds for the Complexity of Multicoloring

Tight Lower Bounds for the Complexity of Multicoloring

Fine-Grained Time Complexity of Constraint Satisfaction Problems

Fine-grained complexity of rainbow coloring and its variants

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