Deterministic Parameterized Algorithms for the Graph Motif Problem

Pinter, Ron Y.; Shachnai, Hadas; Zehavi, Meirav

doi:10.1007/978-3-662-44465-8_50

Cited by 10 publications

(9 citation statements)

References 26 publications

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“…LGM [22]. GM on trees can be solved in n O(c) time where c is the number of colors in M [12], but is W [1]-hard with respect to c [12].…”

Section: General Graphs Treesmentioning

confidence: 99%

Graph Motif Problems Parameterized by Dual

Fertin¹,

Komusiewicz²

2019

Preprint

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Let G = (V, E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G, a multiset M of colors built from C, and asks whether there is a subset S ⊆ V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex v of V may choose its color from a list L(v) ⊆ C of colors.We study the three problems GM, CGM, and LGM, parameterized by the dual parameter ℓ := |V | − |M |. For general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no (2 − ǫ) ℓ • |V | O(1) -time algorithm, which implies that a previous algorithm, running in O(2 ℓ • |E|) time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to ℓ even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in O(3 ℓ • |V |) time but admits no polynomial-size problem kernel, while CGM can be solved in O( √ 2 ℓ + |V |) time and admits a polynomial-size problem kernel.

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“…LGM [22]. GM on trees can be solved in n O(c) time where c is the number of colors in M [12], but is W [1]-hard with respect to c [12].…”

Section: General Graphs Treesmentioning

confidence: 99%

Graph Motif Problems Parameterized by Dual

Fertin¹,

Komusiewicz²

2019

Preprint

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“…It is indeed in FPT when parameterized by the size of the motif. At least seven different papers gave an FPT algorithm [22,4,28,33,5,41,40]. The best (randomized) algorithm runs in time O * (2 k ) where the O * notation suppresses polynomial factors [5,41] and works well in practice for small values of k, even with hundreds of millions of edges [6].…”

Section: Theorem 4 ([9]mentioning

confidence: 99%

“…The best (randomized) algorithm runs in time O * (2 k ) where the O * notation suppresses polynomial factors [5,41] and works well in practice for small values of k, even with hundreds of millions of edges [6]. The current best deterministic algorithm takes time O * (5.22 k ) [40]. However, an algorithm running in time O * ((2 − ǫ) k ) would break the 2 n barrier in solving Set Cover instances with n elements (that is, would disprove SCH) [5].…”

Section: Theorem 4 ([9]mentioning

confidence: 99%

The Graph Motif problem parameterized by the structure of the input graph

Bonnet

Sikora

2017

Discrete Applied Mathematics

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The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. In the many utilizations of Graph Motif, however, the input graph originates from real-life applications and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way. * An extended abstract of this work appears in [10].

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“…Fellows et al [19] isolated the complexity of the problem to the parameter k (the motif size) by presenting a fixedparameter O * (f (k))-time 4 algorithm relying on color coding [1]. A subsequent race to improve the parameter dependency f (k) ensued [2,6,7,26,32,42], with the three most recent contributions consisting of the randomized O * (2.54 k )-time algorithm of Koutis [32], the randomized O * (2 k )-time algorithm of Björklund et al [6,7], and the deterministic O * (5.22 k )-time algorithm of Pinter, Scachnai, and Zehavi [42]. There is complexity-theoretic evidence that algorithms with running time O * ((2 − ) k ) for any constant > 0 do not exist [6,Theorem 6].…”

Section: Motivation and Earliermentioning

confidence: 99%

Engineering Motif Search for Large Graphs

Björklund

Kaski

Kowalik

et al. 2014

2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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In the graph motif problem, we are given as input a vertexcolored graph H (the host graph) and a multiset of colors M (the motif). Our task is to decide whether H has a connected set of vertices whose multiset of colors agrees with M . The graph motif problem is NP-complete but known to admit parameterized algorithms that run in linear time in the size of H. We demonstrate that algorithms based on constrained multilinear sieving are viable in practice, scaling to graphs with hundreds of millions of edges as long as M remains small. Furthermore, our implementation is topologyinvariant relative to the host graph H, meaning only the most crude graph parameters (number of edges and number of vertices) suffice in practice to determine the algorithm performance.

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Deterministic Parameterized Algorithms for the Graph Motif Problem

Cited by 10 publications

References 26 publications

Graph Motif Problems Parameterized by Dual

Graph Motif Problems Parameterized by Dual

The Graph Motif problem parameterized by the structure of the input graph

Engineering Motif Search for Large Graphs

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