Faster Parameterized Algorithms for Deletion to Split Graphs

Ghosh, Esha; Kolay, Sudeshna; Kumar, Manoranjan; Misra, Pranabendu; Panolan, Fahad; Rai, Ashutosh; Ramanujan, M. S.

doi:10.1007/s00453-013-9837-5

Cited by 41 publications

(34 citation statements)

References 18 publications

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“…This implies, in particular, that an n-vertex (r, ℓ)-graph has at most (n + 1) 2rℓ distinct (r, ℓ)-partitions that can be generated in polynomial time if max{r, ℓ} ≤ 2. This result generalizes the fact that a split graph has at most n + 1 split partitions [12], which was used in the algorithms of [11].…”

Section: Introductionsupporting

confidence: 65%

“…Concerning the existence of polynomial kernels for (r, ℓ)-Vertex Deletion, a challenging research avenue is to apply the techniques used by Kratsch and Wahlström [15] for obtaining a randomized polynomial kernel for OCT to the cases (2, 1), (1,2), and (2, 2), or to prove that these problems do not admit polynomial kernels. The ideas for the case (1, 1) may also be helpful [11].…”

Section: Introductionmentioning

confidence: 99%

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Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

et al. 2016

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For two integers r, ℓ ≥ 0, a graph G = (V, E) is an (r, ℓ)graph if V can be partitioned into r independent sets and ℓ cliques. In the parameterized (r, ℓ)-Vertex Deletion problem, given a graph G and an integer k, one has to decide whether at most k vertices can be removed from G to obtain an (r, ℓ)-graph. This problem is NP-hard if r+ℓ ≥ 1 and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of (r, ℓ)-Vertex Deletion was known for all values of (r, ℓ) except for (2, 1), (1, 2), and (2, 2). We prove that each of these three cases is FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of k. We consider as well the version of (r, ℓ)-Vertex Deletion where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem.

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Section: Introductionsupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

et al. 2016

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“…We have presented the first subexponential algorithm for Proper Interval Completion, running in time k O(k 2/3 ) + O(nm(kn + m)). As many algorithms for completion problems in similar graph classes [3,6,8,9] run in time O (k O( √ k) ), it is tempting to ask for such a running time also in our case. The bottleneck in the presented approach is the trade-offs between the two layers of our dynamic programming.…”

Section: Discussionmentioning

confidence: 99%

A Subexponential Parameterized Algorithm for Proper Interval Completion

Bliznets

Fomin

Pilipczuk

et al. 2015

SIAM J. Discrete Math.

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In the Proper Interval Completion problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of Proper Interval Completion from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999], who showed an algorithm for the problem working in O(16time. In this paper we present an algorithm with running time k, which is the first subexponential parameterized algorithm for Proper Interval Completion.

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“…(1) For the problems involving finite forbidden induced subgraphs, improving the kernel obtained from the general methods is an important research direction. Along this line, vertex cover admits an improved kernel with 2k vertices [6] and split vertex deletion admits an improved kernel with O.k 3 / vertices [17] . The question to be considered is whether we can obtain improved kernels for other problems such as cluster vertex deletion, chain vertex deletion, threshold vertex deletion, and co-trivially perfect vertex deletion.…”

Section: Challenges and Further Researchmentioning

confidence: 99%

An overview of kernelization algorithms for graph modification problems

Liu

Wang

Guo

2014

Tinshhua Sci. Technol.

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Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

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Faster Parameterized Algorithms for Deletion to Split Graphs

Cited by 41 publications

References 18 publications

Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

A Subexponential Parameterized Algorithm for Proper Interval Completion

An overview of kernelization algorithms for graph modification problems

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