Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Jansen, Bart M. P.

doi:10.1007/978-3-642-13073-1_18

Cited by 6 publications

(4 citation statements)

References 22 publications

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“…To avoid using weight zero, a similar result can be obtained by assigning weights m and 1 instead of 1 and 0, respectively, where m is the number of edges in G. For this case, Lemma 5.1 implies that an independent set of size t in G corresponds to a spanning arborescence of leaf weight (t + 1)m, and a similar inapproximability result holds, as Jansen [15] proved for the undirected version.…”

Section: Lemma 42 If Algorithmmentioning

confidence: 84%

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Leafy Spanning Arborescences in DAGs

Fernandes¹,

Lintzmayer²

2020

Preprint

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Broadcasting in a computer network is a method of transferring a message to all recipients simultaneously. It is common in this situation to use a tree with many leaves to perform the broadcast, as internal nodes have to forward the messages received, while leaves are only receptors. We consider the subjacent problem of, given a directed graph D, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. In this paper, we concentrate on the class of rooted directed acyclic graphs, for which the problem is known to be MaxSNP-hard. A 2-approximation was previously known for this problem on this class of directed graphs. We improve on this result, presenting a 3 2 -approximation. We also adapt a result for the undirected case and derive an inapproximability result for the vertex-weighted version of Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs.

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Section: Lemma 42 If Algorithmmentioning

confidence: 84%

“…Jansen [15] proved that, unless P = NP, this version of the problem does not admit a polynomialtime ratio O(n The Independent Set problem consists of the following: given a graph G, find an independent set in G with as many vertices as possible.…”

Section: Inapproximability Of the Vertex-weighted Versionmentioning

confidence: 99%

Leafy Spanning Arborescences in DAGs

Fernandes¹,

Lintzmayer²

2020

Preprint

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“…Then, rETH states that there is a constant c > 0 such that there is no randomized algorithm that decides 3-SAT in time O * (2 c•n ) with (two-sided) error probability at most 1 3 (Dell et al 2014). Using the sparsification lemma (Jansen 2010) one can show the following, see Exercises 14.1 of (Cygan et al 2015a). Theorem 0.11.…”

Section: Hardnessmentioning

confidence: 99%

Parameterized Approximation Algorithms for K-center Clustering and Variants

Bandyapadhyay

Friggstad

Mousavi

2022

AAAI

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k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^{O((klog k)/{epsilon})}dn-time (1+epsilon)-approximation for Euclidean k-center, where d is the dimension. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} log k)})-time (1+epsilon)-approximation. In particular, the running time is roughly O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)}) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^{O(klog k)}n^2 time 3-approximation for NUkC, and a 2^{O((klog k)/epsilon)}dn time (1+\epsilon)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

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“…Then, rETH states that there is a constant c > 0 such that there is no randomized algorithm that decides 3-SAT in time O * (2 c•n ) with (two-sided) error probability at most 1 3 [DHM + 14]. Using the sparsification lemma [Jan10] one can show the following, see Exercises 14.1 of [CFK + 15a].…”

Section: Hardnessmentioning

confidence: 99%

Parameterized Approximation Algorithms for $k$-Center Clustering and Variants

Bandyapadhyay¹,

Friggstad²,

Mousavi³

2021

Preprint

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k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2 O((k log k)/ε) dn-time (1 + ε)approximation for Euclidean k-center, where d is the dimension.We give a faster algorithm for small dimensions: roughly speaking an) in the plane. We complement our algorithmic result with a matching hardness lower bound.We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2 O(k log k) n 2 time 3-approximation for NUkC in general metrics, and a 2 O((k log k)/ε) dn time (1 + ε)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

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Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Cited by 6 publications

References 22 publications

Leafy Spanning Arborescences in DAGs

Leafy Spanning Arborescences in DAGs

Parameterized Approximation Algorithms for K-center Clustering and Variants

Parameterized Approximation Algorithms for $k$-Center Clustering and Variants

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