Parameterized Power Vertex Cover

Angel, Éric; Bampis, Evripidis; Escoffier, Bruno; Lampis, Michael

doi:10.1007/978-3-662-53536-3_9

Cited by 11 publications

(35 citation statements)

References 16 publications

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“…By the definition of tree-depth, after removal of 2 √ n vertices from G, the maximum tree-depth of each resulting disconnected component is log(6 · c √ n ) = √ n · log(c) + log (6) . Our algorithm makes use of a technique introduced in [23] (see also [1,21]) for approximating problems that are W-hard by treewidth. If the hardness of the problem arises from the need of the dynamic programming table to store tw large numbers (in our case, the distances of the vertices in the bag from the closest selection), we can significantly speed up the algorithm by replacing all values by the closest integer power of (1 + δ), for some appropriately chosen δ, thus reducing the table size from d tw to (log (1+δ) d) tw .…”

Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

Structurally parameterized d-scattered set

Katsikarelis

Lampis

Paschos

2022

Discrete Applied Mathematics

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In d-Scattered Set we are given an (edge-weighted) graph and are asked to select at least k vertices, so that the distance between any pair is at least d, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following:• For any d ≥ 2, an O * (d tw )-time algorithm, where tw is the treewidth of the input graph and a tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set.• d-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if k is an additional parameter.• A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness.• A 2 O(td 2 ) -time algorithm parameterized by tree-depth (td), as well as a matching ETHbased lower bound, both for unweighted graphs.We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter > 0, runs in time O * ((tw/ ) O(tw) ) and returns a d/(1+ )-scattered set of size k, if a d-scattered set of the same size exists. arXiv:1709.02180v4 [cs.CC] 10 Nov 2018Our contribution: First, in Section 3 we present a lower bound of (d − ) tw · n O(1) on the complexity of any algorithm solving d-Scattered Set parameterized by tw, based on the Strong Exponential Time Hypothesis (SETH [19,20]). This result can be seen as a non-trivial extension of the bound of (2 − ) tw · n O(1) for Independent Set ([24]) for larger values of d, for which the construction is required to be much more compact in terms of encoded information per unit of treewidth. Next, in Section 4 we provide a dynamic programming algorithm of running time O * (d tw ), matching this lower bound, over a given tree decomposition of width tw. The algorithm actually solves the counting version of d-Scattered Set, making use of standard techniques (dynamic programming on tree decompositions), with an application of the fast subset convolution technique of [2] (or state changes [7,30]) to bring the running time down to match the size of the dynamic programming tables.Having thus identified the complexity of the problem with respect to tw, we next focus on the more restrictive parameters vc and fvs and we show in Section 5 that the edge-weighted d-Scattered Set problem parameterized by vc + k is W[1]-hard. If, on the other hand, all edge-weights are set to 1, then d-Scattered Set (the unweighted variant) parameterized by fvs + k is W[1]-hard. Our reductions also imply lower bounds based on the Exponential Time Hypothesis (ETH [19,20]), yet we do not believe these to be tight, due to the quadratic increase in parameter size (as the construction's focus lies on the edges). One observation we can make is that there are few cases where we can expect to obtain an FPT algorithm without bounding the va...

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Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

Structurally parameterized d-scattered set

Katsikarelis

Lampis

Paschos

2022

Discrete Applied Mathematics

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“…Covering Problems parameterized by Graph Width Parameters. Several works in literature also study the approximability of variants of VERTEX COVER and DOMINATING SET parameterized by graph widths [105,135]. These variants include:…”

Section: Connected Vertex Covermentioning

confidence: 99%

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

et al. 2020

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“…Our first approximation algorithm, which is an approximation scheme for the optimal value of ∆ * , relies on a method introduced in [36] (see also [3]), and a theorem of [11]. The high-level idea is the following: intuitively, the obstacle that stops us from obtaining an FPT running time with the dynamic programming algorithm of Theorem 19 is that the dynamic program is forced to store some potentially large values for each vertex.…”

Section: Approximation Algorithmsmentioning

confidence: 99%

Parameterized (Approximate) Defective Coloring

Belmonte¹,

Lampis²,

Mitsou³

2020

SIAM J. Discrete Math.

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In Defective Coloring we are given a graph G = (V, E) and two integers χ d , ∆ * and are asked if we can partition V into χ d color classes, so that each class induces a graph of maximum degree ∆ * . We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if χ d = 2. As expected, this hardness can be extended to larger values of χ d for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any χ d = 2, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETHbased lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in n o(pw) , essentially matching the complexity of an algorithm obtained with standard techniques.We complement these results by considering the problem's approximability and show that, with respect to ∆ * , the problem admits an algorithm which for any > 0 runs in time (tw/ ) O(tw) and returns a solution with exactly the desired number of colors that approximates the optimal ∆ * within (1 + ). We also give a (tw) O(tw) algorithm which achieves the desired ∆ * exactly while 2-approximating the minimum value of χ d . We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to χ d , even when an extra constant additive error is also allowed.

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Parameterized Power Vertex Cover

Cited by 11 publications

References 16 publications

Structurally parameterized d-scattered set

Structurally parameterized d-scattered set

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterized (Approximate) Defective Coloring

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