A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

Bazgan, Cristina; Nichterlein, André; Niedermeier, Rolf

doi:10.1007/978-3-319-18173-8_3

Cited by 12 publications

(11 citation statements)

References 21 publications

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“…Our first application of fractalism is the Length-Bounded Edge-Cut problem [2], also known as the problem of finding bounded edge undirected cuts [27], or the Shortest Path Most Vital Edges problem [4,36].…”

Section: Length-bounded Edge-cutmentioning

confidence: 99%

“…Herein, an undirected graph G = (V, E) with s, t ∈ V , and two integers k, ℓ ∈ N are given, and the question is whether it is possible to delete at most k edges such that the shortest s-t path is of length at least ℓ. Using a simple branching algorithm, one can show that LBEC(k, ℓ) is fixed-parameter tractable for the combined parameter (k, ℓ) [4,27]. Golovach and Thilikos [27] posed as an open problem whether LBEC(k, ℓ) admits a polynomial kernel.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Length-bounded Edge-cutmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractals for Kernelization Lower Bounds

Fluschnik¹,

Hermelin²,

Nichterlein³

et al. 2018

SIAM J. Discrete Math.

Self Cite

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“…In contrast to many other cut problems on graphs (e.g., multiway cut, multicut, sparsest cut, balanced cut, maximum cut, multiroute cut), the known approximations of the minimum L-bounded cut problem are substantially weaker. In this work we focus on algorithms for restricted graph classes, namely planar graphs, bounded genus graphs and graphs with bounded, yet not constant, treewidth, and provide new results for the L-bounded cut problem on them; the results for planar graphs solve one of the open problems suggested by Bazgan et al [BNN15]. We also remark that the generic approximation scheme of Czumaj et.…”

mentioning

confidence: 94%

“…From the point of view of parameterized complexity, the problem was also studied by Golovach and Thilikos [GT11], Bazgan et al [BNN15] and by Fluschnik et al [FHNN16].…”

mentioning

confidence: 99%

On Algorithms Employing Treewidth for $L$-bounded Cut Problems

Kolman¹

2018

JGAA

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Given a graph G = (V, E) with two distinguished vertices s, t ∈ V and an integer parameter L > 0, an L-bounded cut is a subset F of edges (vertices) such that the every path between s and t in G \ F has length more than L. The task is to find an L-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70's, it is not much understood yet. The problem is known to be N P-hard to approximate within a small constant factor even for L ≥ 4 (for L ≥ 5 for the vertex cuts). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only O(n 2/3 ) in the edge case, and O( √ n) in the vertex case, where n denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edgeand the vertex-version of the problem optimally in time O(L 3L n). That is, the problem is fixed parameter tractable (FPT) with respect to L on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex version of the problem. We describe an algorithm that for a given a graph G, its tree decomposition of treewidth τ and vertices s and t computes a τ -approximation of the minimum L-bounded s − t vertex cut; if the decomposition is not given, then the approximation ratio is O(τ √ log τ ). For graphs with treewidth bounded by O(n 1/2−ǫ ) for any ǫ > 0, but not by a constant, this is the best approximation in terms of n that we are aware of. IntroductionThe subject of this paper is a variation of the classical s-t cut problem, namely the minimum L-bounded edge (vertex) cut problem: given a graph G = (V, E) with two distinguished vertices s, t ∈ V and an integer parameter L > 0, find a subset F of edges (vertices) of minimum cardinality such that every path between s and t in G \ F has length more than L. ⋆

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“…The name derives from the notion of most vital arcs in a networkthose whose deletion decreases the flow or increases the length of the shortest path as much as possible. While the graph problems are well studied [1,2,4,5,16,18,23,27], to our knowledge, geometric versions of locating "most vital" facilities have not been explored. Throughout the paper, the segment barriers will be called simply barriers.…”

Section: Introductionmentioning

confidence: 99%

Most Vital Segment Barriers

Kostitsyna

Löffler

Polishchuk

et al. 2019

Lecture Notes in Computer Science

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We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases. * TU

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A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

Cited by 12 publications

References 21 publications

Fractals for Kernelization Lower Bounds

Fractals for Kernelization Lower Bounds

On Algorithms Employing Treewidth for $L$-bounded Cut Problems

Most Vital Segment Barriers

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