Fast FAST

Alon, Noga; Lokshtanov, Daniel; Saurabh, Saket

doi:10.1007/978-3-642-02927-1_6

Cited by 81 publications

(191 citation statements)

References 22 publications

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“…Ghosh et al [15] showed that Split Completion can be solved in time 2 O( √ k log k) · poly(n) using the framework of Alon, Lokshtanov and Saurabh [1]. However, the following observation immediately yields a very simple combinatorial argument for the existence of such an algorithm.…”

Section: Lemma 53 (Few Split Partitions)mentioning

confidence: 98%

On the Threshold of Intractability

Drange

Dregi

Lokshtanov

et al. 2015

Algorithms - ESA 2015

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We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-hard, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1): [109][110][111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128] 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in 2 O( √ k log k) + poly(n) time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems.

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Section: Lemma 53 (Few Split Partitions)mentioning

confidence: 98%

On the Threshold of Intractability

Drange

Dregi

Lokshtanov

et al. 2015

Algorithms - ESA 2015

Self Cite

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“…FAST is NP-complete [1] but fixed-parameter tractable with respect to k [2,14,16,20,27,42]. The running time of the current best fixed-parameter algorithm is 2 c· √ k + n O(1) where c ≤ 5.24 [20].…”

Section: Problem 51 (Feedback Arc Set In Tournaments (Fast))mentioning

confidence: 99%

“…Observe in this context that Lemma 3.2 is also correct if the input graphs are directed and if a solution contains arc reversals, since we observed arc reversals and deletions to be equivalent in the context of FAST. 2 …”

Section: Problem 51 (Feedback Arc Set In Tournaments (Fast))mentioning

confidence: 99%

Parameterizing Edge Modification Problems Above Lower Bounds

Bevern

Froese

Komusiewicz

2016

Computer Science – Theory and Applications

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We study the parameterized complexity of a variant of the F-free Editing problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing H of induced subgraphs of G, which provides a lower bound h(H) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter := k − h(H), that is, the number of edge modifications above the lower bound h(H). We develop a framework of generic data reduction rules to show fixed-parameter tractability with respect to for K 3 -Free Editing, Feedback Arc Set in Tournaments, and Cluster Editing when the packing H contains subgraphs with bounded solution size. For K 3 -Free Editing, we also prove NP-hardness in case of edge-disjoint packings of K 3 s and = 0, while for K q -Free Editing and q ≥ 6, NP-hardness for = 0 even holds for vertex-disjoint packings of K q s. In addition, we provide NP-hardness results for F-free Vertex Deletion, were the aim is to delete a minimum number of vertices to make the input graph F-free.

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“…This result was later improved by Dujmović, Fernau, and Kaufmann [6] who gave an algorithm with running time O(1.4656 k + kn 2 ). Fernau et al [8] reduced this problem to weighted FAST (feedback arc sets in tournaments) and, using the algorithm of Alon, Lokshtanov, and Saurabh [2] for weighted FAST, gave a subexponential time algorithm that runs in 2 O( √ k log k) + n O (1) time. This reduction also gave a PTAS using the algorithm of Kenyon-Mathieu and Schudy [13].…”

Section: Oscm (One-sided Crossing Minimization)mentioning

confidence: 99%

“…The algorithm in [5] involves several reduction rules for kernelization and the improvement in [6] is obtained by introduction of additional reduction rules which entail more involved analysis. The algorithm in [8] relies on the algorithm in [2] for the more general problem of FAST. Our result suggests that OSCM is significantly easier than FAST in that it does not require any advanced algorithmic techniques or sophisticated combinatorial structures used in the algorithm of [2] for FAST, in deriving a subexponential algorithm.…”

Section: Theorem 2 There Is Nomentioning

confidence: 99%

A Fast and Simple Subexponential Fixed Parameter Algorithm for One-Sided Crossing Minimization

Kobayashi

Tamaki

2014

Algorithmica

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We give a subexponential fixed parameter algorithm for one-sided crossing minimization. It runs in O(k2where n is the number of vertices of the given graph and parameter k is the number of crossings. The exponent of O( √ k) in this bound is asymptotically optimal assuming the Exponential Time Hypothesis and the previously best known algorithm runs in 2 O( √ k log k) + n O(1) time. We achieve this significant improvement by the use of a certain interval graph naturally associated with the problem instance and a simple dynamic program on this interval graph. The linear dependency on n is also achieved through the use of this interval graph.

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Fast FAST

Cited by 81 publications

References 22 publications

On the Threshold of Intractability

On the Threshold of Intractability

Parameterizing Edge Modification Problems Above Lower Bounds

A Fast and Simple Subexponential Fixed Parameter Algorithm for One-Sided Crossing Minimization

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