Mim-Width II. The Feedback Vertex Set Problem

Jaffke, Lars; Kwon, O-joung; Telle, Jan Arne

doi:10.1007/s00453-019-00607-3

“…In particular, Jaffke, Kwon, Strømme and Telle [34] proved that the distance versions of LC‐VSVP problems can be solved in polynomial time for graph classes where mim‐width is bounded and quickly computable. Jaffke, Kwon and Telle [35,36] proved similar results for L ongest I nduced P ath , I nduced D isjoint P aths ,

H

‐I nduced T opological M inor and F eedback V ertex S et. The latter result has recently been generalized to S ubset F eedback V ertex S et and N ode M ultiway C ut , by Bergougnoux, Papadopoulos and Telle [3].…”

Section: Introductionmentioning

confidence: 59%

Bounding the mim‐width of hereditary graph classes

Brettell

¹

,

Horsfield

²

,

Munaro

³

et al. 2021

Journal of Graph Theory

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A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H1, H2)-free graphs. We identify several general classes of (H1, H2)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H1, H2)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width.Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H1, H2)-free graphs. In particular, we classify the mim-width of (H1, H2)-free graphs for all pairs (H1, H2) with |V (H1)| + |V (H2)| ≤ 8. When H1 and H2 are connected graphs, we classify all pairs (H1, H2) except for one remaining infinite family and a few isolated cases.

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“…We resolve in the affirmative the question raised by Jaffke et al [29], also mentioned in [37,38], asking whether there is an XP-time algorithm for SFVS parameterized by the mim-width of a given decomposition. For rank-width and Q-rank-width we provide the first explicit FPT-algorithms with low exponential dependency that avoid the MSO 1 formulation.…”

Section: Our Resultsmentioning

confidence: 77%

“…Recently, Bergougnoux and Kanté [3] extended the uses of this notion to acyclic and connected variants of (σ, ρ) generalized domination and similar problems like FVS. An earlier XP algorithm for FVS parameterized by mim-width had been given by Jaffke et al [29] but instead of the d-neighbor equivalences this algorithm was based on the notions of reduced forests and minimal vertex covers.…”

Section: Our Approachmentioning

confidence: 99%

See 1 more Smart Citation

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

Bergougnoux

¹

,

Papadopoulos

²

,

Telle

³

2022

Self Cite

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The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

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“…Papadopoulos and Tzimas [31,32] proved that Subset Feedback Vertex Set is polynomial-time solvable for sP 1 -free graphs for any s ≥ 1, co-bipartite graphs, interval graphs and permutation graphs, and thus P 4 -free graphs. Some of these results were generalized by Bergougnoux et al [2], who solved an open problem of Jake et al [22] by giving an n O(w 2 ) -time algorithm for Subset Feedback Vertex Set given a graph and a decomposition of this graph of mim-width w. This does not lead to new results for H-free graphs: a class of H-free graphs has bounded mim-width if and only if H ⊆ i P 4 [7].…”

Section: And Np-complete Ifmentioning

confidence: 96%

Computing subset transversals in H-free graphs

Brettell

¹

,

Johnson

²

,

Paesani

³

et al. 2022

Theoretical Computer Science

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We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to Hfree graphs, that is, to graphs that do not contain some xed graph H as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on H-free graphs for every graph H except when H = sP1 + P4 for some s ≥ 1. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for (sP1 + P4)-free graphs for every s ≥ 1.

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Mim-Width II. The Feedback Vertex Set Problem

Cited by 26 publications

References 30 publications

Bounding the mim‐width of hereditary graph classes

Bounding the mim‐width of hereditary graph classes

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

Computing subset transversals in H-free graphs

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