Parameterized Approximation Schemes Using Graph Widths

Lampis, Michael

doi:10.1007/978-3-662-43948-7_64

Cited by 18 publications

(40 citation statements)

References 36 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%

“…The rounding technique as applied in [23] employs randomization and an extensive analysis to procure the bounds on the propagation of error, while we only require a deterministic adaptation of the rounding process without making use of the advanced machinery there introduced, as for our particular case, the bound on the rounding error can be straightforwardly obtained. The main tool we require is the following definition of an addition-rounding operation, denoted by ⊕: for two non-negative numbers x 1 , x 2 , we define x 1 ⊕ x 2 := 0, if x 1 = x 2 = 0.…”

Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

“…As a final note, observe that due to the use of the · function in the definition of our ⊕ operator, all our values will be rounded down, in contrast to the original version of the technique (from [23]), where depending on a randomly chosen number ρ, the values could be rounded either down or up. This means there will be some value x, such that x ⊕ 1 = x, or (1 + δ) x = (1 + δ) log (1+δ) (x+1) (we would have x ≈ 1/δ).…”

Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

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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%

Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

Section: Tree-depth: Tight Eth Lower Boundmentioning

confidence: 99%

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Structurally parameterized d-scattered set

Katsikarelis

Lampis

Paschos

2022

Discrete Applied Mathematics

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“…Our algorithm will rely heavily on a technique introduced in [35] to approximate problems which are W-hard by treewidth (see also [3], as well as [11] for a similar approach). The idea is that, if the hardness of the problem is due to the fact that the DP table needs to store tw large numbers (in our case, the distances of the vertices in the bag from the closest center), we can significantly speed up the algorithm if we replace all entries by the closest integer power of (1 + δ), for some appropriately chosen δ.…”

Section: Treewidth: Fpt Approximation Schemementioning

confidence: 99%

“…The tw approximation algorithm is based on a technique introduced in [35], while the cw algorithm relies on a new extension of an idea from [27], which may be of independent interest. Thanks to these approximation algorithms, we arrive at an improved understanding of the complexity of (k, r)-Center by including the question of approximation, and obtain algorithms which continue to work efficiently even for large values of r. Figure 1 illustrates the relationships between parameters and Related Work: Our work follows upon recent work by [13], which showed that (k, r)-Center can be solved in O * ((2r + 1) tw ), but not faster (under SETH), while its connected variant can be solved in O * ((2r + 2) tw ), but not faster.…”

Section: Introductionmentioning

confidence: 99%

Structural parameters, tight bounds, and approximation for (k,r)-center

Katsikarelis

Lampis

Paschos

2019

Discrete Applied Mathematics

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In (k, r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically:• For any r ≥ 1, we show an algorithm that solves the problem in O * ((3r + 1) cw ) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r = 1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.• We strengthen previously known FPT lower bounds, by showing that (k, r)-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.• We show that the complexity of the problem parameterized by tree-depth is 2 Θ(td 2 ) , by showing an algorithm of this complexity and a tight ETH-based lower bound.We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k, r. In particular, we give algorithms which, for any > 0, run in time O * ((tw/ ) O(tw) ), O * ((cw/ ) O(cw) ) and return a (k, (1 + )r)-center if a (k, r)-center exists, thus circumventing the problem's W-hardness.

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Structurally Parameterized d-Scattered Set

Katsikarelis¹,

Lampis²,

Paschos³

2018

Graph-Theoretic Concepts in Computer Science

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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. IntroductionIn this paper we study the d-Scattered Set problem: given graph G = (V, E) and a metric weight function w : E → N + that gives the length of each edge, we are asked if there exists a set K of at least k selections from V , such that the distance between any pair v, u ∈ K is at least d(v, u) ≥ d, where d(v, u) denotes the shortest-path distance from v to u under weight function w. If w assigns weight 1 to all edges, the variant is called unweighted.The problem can already be seen to be hard, as it generalizes Independent Set (for d = 2), even to approximate (under standard complexity assumptions), i.e. the optimal k cannot be approximated to n 1− in polynomial time [19], while an alternative name is 28,13]. This hardness prompts the analysis of the problem when the input graph is of restricted structure, our aim being to provide a comprehensive account of the complexity of d-Scattered Set through various upper and lower bound results. Our viewpoint is parameterized: we consider the well-known structural parameters treewidth tw, tree-depth td, vertex cover number vc and feedback vertex set number fvs, that comprehensively express the intended restrictions on the input graph's structure (as they range in size and applicability), while we examine both the edge-weighted and unweighted variants of the problem. 1

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Parameterized Approximation Schemes Using Graph Widths

Cited by 18 publications

References 36 publications

Structurally parameterized d-scattered set

Structurally parameterized d-scattered set

Structural parameters, tight bounds, and approximation for (k,r)-center

Structurally Parameterized d-Scattered Set

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