On Feedback Vertex Set: New Measure and New Structures

Cao, Yixin; Chen, Jianer; Liu, Yang

doi:10.1007/s00453-014-9904-6

Cited by 58 publications

(59 citation statements)

References 25 publications

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“…-Third, A constructs an auxiliary graph G = (V , E ) by setting V to be the set of equivalence classes of ∼ and E to contain an edge between A, B ∈ V iff there exist vertices a ∈ A, b ∈ B such that ab ∈ E(G). -Finally, A tries to find a feedback vertex set in G of size at most j in time O(3.83 j · j|V | 2 ) [7]. If no such feedback vertex set exists, then A terminates the given branch; otherwise it adds the feedback vertex set to S and outputs S.…”

Section: Lemma 11 It Is Possible To Compute a K-well-structured Modulmentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such wellstructured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

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Section: Lemma 11 It Is Possible To Compute a K-well-structured Modulmentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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“…Given a graph G and an integer k as input, the Feedback Vertex Set problem asks whether G has a vertex subset of size at most k whose removal makes it a forest, which is a graph without cycles. The Feedback Vertex Set problem is known to admit an FPT algorithm [2,12] and the running time has been subsequently improved by a series of papers [24,17,15,11,5,3,8,20]. Also, Thomassé [27] showed that it admits a kernel on Opk 2 q vertices.…”

Section: Introductionmentioning

confidence: 99%

A Polynomial Kernel for Block Graph Deletion

Kim

Kwon

2017

Algorithmica

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Abstract. In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with Opk 6 q vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of 'complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10k¨nOp1q .

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“…Now that we have stored the entire graph, use any one of the various known FPT algorithms [8,23] to solve the F V S(k) problem.…”

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confidence: 99%

Parameterized Streaming: Maximal Matching and Vertex Cover

Chitnis¹,

Cormode²,

Hajiaghayi³

et al. 2014

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

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As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams.In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems. Here, we are given a parameter k and the objective is to decide whether there is a solution bounded by k. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of Maximal Matching and Vertex Cover. We consider various models for a graph stream on n nodes: the insertion-only model where the edges can only be added, and the dynamic model where edges can be both inserted and deleted. More formally, we show the following results:• In the insertion only model, there is a one-pass deterministic algorithm for the parameterized Vertex Cover problem which computes a sketch using

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On Feedback Vertex Set: New Measure and New Structures

Cited by 58 publications

References 25 publications

Solving Problems on Graphs of High Rank-Width

Solving Problems on Graphs of High Rank-Width

A Polynomial Kernel for Block Graph Deletion

Parameterized Streaming: Maximal Matching and Vertex Cover

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