Subset Feedback Vertex Set in Chordal and Split Graphs

Philip, Geevarghese; Rajan, Varun; Saurabh, Saket; Tale, Prafullkumar

doi:10.1007/s00453-019-00590-9

Cited by 10 publications

(4 citation statements)

References 22 publications

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“…Notable differences between the two latter problems regarding their complexity status is the class of split graphs and 4P 1 -free graphs for which Subset Feedback Vertex Set is NP-complete [17,36], as opposed to the Feedback Vertex Set problem [11,39,7]. Inspired by the NP-completeness on chordal graphs, Subset Feedback Vertex Set restricted on (subclasses of) chordal graphs has attracted several researchers to obtain fast, still exponential-time, algorithms [22,37].…”

Section: Introductionmentioning

confidence: 99%

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2022

Lecture Notes in Computer Science

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Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph G = (V, E) and a set S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains NP-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the leafage that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ, we provide an algorithm for SFVS with running time n O(ℓ) , thus improving upon n O(ℓ 2 ) , the running time of the previously known algorithm obtained for graphs with bounded mim-width. We complement our result by showing that SFVS is W[1]-hard parameterized by ℓ. Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains NP-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs of mim-width one, which is faster than the previously known algorithm obtained for graphs with bounded mim-width.

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Section: Introductionmentioning

confidence: 99%

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2022

Lecture Notes in Computer Science

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“…However, designing a faster algorithm for SFVS in Chordal Graphs seems challenging. Only recently did Philip et al [25] improve the running bound to O * (2 k ), where they needed to consider many cases of the clique-tree structures of the chordal graphs. For some cases, they needed to branch into 7 branches.…”

Section: Introductionmentioning

confidence: 99%

“…[25]) Let G be a chordal graph and T ⊆ V be the terminal set. A vertex set S ⊆ V is a subset feedback vertex set of G if and only if G − S contains no T -triangles.…”

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confidence: 99%

Breaking the Barrier 2k for Subset Feedback Vertex Set in Chordal Graphs

Bai

Xiao

2022

Preprint

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The Subset Feedback Vertex Set problem (SFVS), to delete $k$ vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains $|NP|$-hard even in split and chordal graphs, and SFVS in Chordal Graphs can be considered as a special case of the 3-Hitting Set problem. However, it is not easy to solve SFVS in Chordal Graphs faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS in Chordal Graphs can be solved in $2^{k}n^{\mathcal{O}(1)}$ time, slightly improving the best result $2.076^{k}n^{\mathcal{O}(1)}$ for 3-Hitting Set. In this paper, we break the ``$2^{k}$-barrier'' for SFVS in Chordal Graphs by giving a $1.619^{k}n^{\mathcal{O}(1)}$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.

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“…Notable differences between the two later problems regarding their complexity status is the class of split graphs and 4P 1 -free graphs for which Subset Feedback Vertex Set is NP-complete [14,29], as opposed to the Feedback Vertex Set problem [12,32,8]. Inspired by the NP-completeness on chordal graphs, Subset Feedback Vertex Set restricted on (subclasses of) chordal graphs has attracted several researchers to obtain faster, still exponential-time, algorithms [18,30].…”

Section: Introductionmentioning

confidence: 99%

Computing Subset Feedback Vertex Set via Leafage

Papadopoulos¹,

Tzimas²

2021

Preprint

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Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the Subset Feedback Vertex Set (SFVS) problem: given a graph G = (V, E) and a set S ⊆ V , SFVS asks for a minimum set of vertices that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains NP-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the leafage that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage , we provide an algorithm for SFVS with running time n O( ) . Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the smallest number among the maximum number of leaves of all subtrees in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains NP-complete on undirected path graphs, i.e., graphs having vertex leafage at most two. Moreover, we strengthen previously-known polynomial-time algorithm for SFVS on directed path graphs that form a proper subclass of undirected path graphs and graphs of mim-width one.

show abstract

Subset Feedback Vertex Set in Chordal and Split Graphs

Cited by 10 publications

References 22 publications

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Breaking the Barrier 2k for Subset Feedback Vertex Set in Chordal Graphs

Computing Subset Feedback Vertex Set via Leafage

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