Parameterized Complexity of Length-bounded Cuts and Multicuts

Knop, Dušan; Dvořák, Pavel

doi:10.1007/s00453-018-0408-7

Cited by 14 publications

(10 citation statements)

References 23 publications

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“…Lastly, we showed that Length-Bounded Cut is W [1]-hard with respect to the combined parameter pathwidth and maximum degree. This combines two results by Dvořák and Knop [9] and Bazgan et al [3]. It…”

Section: Discussionsupporting

confidence: 83%

“…We confirm this conjecture by showing a dynamic-programming based polynomial-time algorithm. We strengthen the W[1]-hardness result of Dvořák and Knop [9]. Our reduction is shorter, seems simpler to describe, and the target of the reduction has stronger structural properties.…”

supporting

confidence: 66%

“…Last but not least, we show in Theorem 1 that Length-Bounded Cut is W [1]-hard for the combined parameter pathwidth and maximum degree of the input graph G. This is a nontrivial strengthening of the reduction provided by Dvořák and Knop [9], where the degree cannot be bounded by a function of the parameter. Furthermore, our reduction implies that assuming ETH, there is no f (k) • n o(k) -time algorithm for Length-Bounded Cut, where k is pathwidth of the input graph (whereas the reduction of Dvořák and Knop refutes only f (k) • n √ k -time algorithms).…”

Section: Our Contributionmentioning

confidence: 84%

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Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Bentert¹,

Heeger²,

Knop³

2019

Preprint

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In the presented paper we study the Length-Bounded Cut problem for special graph classes as well as from a parameterized-complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set of edges F of size at most β such that every s-t-path of length at most λ in G contains some edge in F . Bazgan et al. [3] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph G is a proper interval graph. We confirm this conjecture by showing a dynamic-programming based polynomial-time algorithm. We strengthen the W[1]-hardness result of Dvořák and Knop [9]. Our reduction is shorter, seems simpler to describe, and the target of the reduction has stronger structural properties. Consequently, we give W[1]hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.these problems have been introduced by Adámek and Koubek [1] and the Length-bounded Cut problem is formally defined as follows. Input:An undirected graph G = (V, E), two vertices s, t, and two positive integers β, λ. Task:Decide whether there exists a subset F ⊆ E with |F | ≤ β such that there is no s-t-path in G − F of length at most λ. Length-Bounded CutIf in the above definition one plugs-in λ = |G|, then one is left with the Edge Cut problem; a polynomial-time-solvable problem. However, Baier et al. [2] showed that the Length-Bounded Cut problem is NP-hard already for λ = 4. On the other hand, the related Length-Bounded Flow problem, where we restrict the flow to paths of length at most λ, can be solved in polynomial time via a reduction to linear programming [2,18,16]. Before we give an overview of our results, we discuss the related work with the focus on parameterized algorithms and algorithms for special graph classes. Related WorkNote that the result of Baier et al.[2] in fact gives para-NP-hardness for Length-Bounded Cut for the parameter λ. Thus, in order to obtain tractability results one has to either consider a different parameterization or combine λ with some other parameter. The first study of Length-Bounded Cut from the viewpoint of parameterized complexity was done by Golovach and Thilikos [12]. They showed that Length-Bounded Cut is in FPT for the combined parameter β + λ. It is worth noting that parameterization by β only leads to para-NP-hardness as well [17,21]. Later, Fluschnik et al. [10] proved that it is unlikely that a polynomial kernel in β + λ exists. Dvořák and Knop [9] considered structural parameters for the Length-Bounded Cut problem. They showed that it is W[1]-hard when parameterized by the pathwidth of the input graph while it is fixed-parameter tractable when parameterized by treedpeth on the input graph.It is worth pointing out that Length-Bounded Cut is one of just a few problems with such a parameterized dichotomy. Kolman [15] gave an O (λ τ • |G|)-time algorithm f...

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Section: Discussionsupporting

confidence: 83%

supporting

confidence: 66%

Section: Our Contributionmentioning

confidence: 84%

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Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Bentert¹,

Heeger²,

Knop³

2019

Preprint

Self Cite

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“…The study of structural parameters which trade off the generality of treewidth for improved algorithmic properties is by now a standard topic in parameterized complexity. The most common type of work here is to consider a problem that is intractable parameterized by treewidth and see whether it becomes tractable parameterized by vertex cover or treedepth [2,10,13,16,17,31,32,35,34,36,42,41]. See [1] for a survey of results of this type.…”

Section: Related Workmentioning

confidence: 99%

Fine-grained Meta-Theorems for Vertex Integrity

Lampis¹,

Mitsou²

2021

Preprint

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Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph G of vertex integrity k and an FO formula φ with q quantifiers, deciding if G satisfies φ can be done in time 2 O(k 2 q+q log q) + n O(1) ;(ii) for MSO formulas with q quantifiers, the same can be done in time 2 2 O(k 2 +kq)+ n O(1) . Both results are obtained using kernelization arguments, which pre-process the input to sizes 2 O(k 2 ) q and 2 O(k 2 +kq) respectively.The complexities of our meta-theorems are significantly better than the corresponding metatheorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly 2 O(kq) and 2 2 O (k+q) complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on k is best possible. More precisely, we show that it is not possible to decide FO formulas with q quantifiers in time 2 o(k 2 q) , and that there exists a constant-size MSO formula which cannot be decided in time 2 2 o(k 2 ), both under the ETH. Hence, the quadratic blow-up in the dependence on k is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.

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“…The directed variant with positive edge lengths remains NP-hard on planar graphs where the source and the sink vertex are incident to the same face [41]. Recently, Dvořák and Knop [21] showed that the problem can be solved in polynomial time on graphs of bounded treewidth. Here, we answer an open question [27] concerning the existence of a polynomial kernel with respect to the combined parameter (k, ℓ).…”

Section: Length-bounded Edge-cut (Lbec) Inputmentioning

confidence: 99%

Fractals for Kernelization Lower Bounds

Fluschnik¹,

Hermelin²,

Nichterlein³

et al. 2018

SIAM J. Discrete Math.

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The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim. 2011], we show that, unless NP ⊆ coNP / poly, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most k edges such that the resulting graph has no s-t path of length shorter than ℓ) parameterized by the combination of k and ℓ has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems. Along the way, we show that LBEC remains NP-hard on planar graphs, a result which we believe is interesting in its own right.

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Parameterized Complexity of Length-bounded Cuts and Multicuts

Cited by 14 publications

References 23 publications

Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Fine-grained Meta-Theorems for Vertex Integrity

Fractals for Kernelization Lower Bounds

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