On the Approximability of Dense Steiner Tree Problems

Hauptmann, Mathias

doi:10.1142/9789812770998_0006

Cited by 4 publications

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“…In particular, there exist many generalizations of the Steiner tree problem, so these may be a natural starting point to attempt to generalize the results. A helpful overview of Steiner-type problems can be found in [20]. Regarding the DMST, it may be difficult to generalize a similar approach to closely related problems.…”

Section: Discussion and Future Workmentioning

confidence: 99%

Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms

et al. 2023

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We present two algorithms in the quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability: one for producing an approximately optimal Steiner tree, and one for producing an exact directed minimum spanning tree, each of which uses O˜(n1/4) rounds of communication and O˜(n9/4) messages, achieving a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. Additionally, we characterize the constants and logarithmic factors involved in our algorithms as well as related classical algorithms, otherwise obscured by O˜ notation, revealing that advances are needed to render both the quantum and classical algorithms practical.

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Section: Discussion and Future Workmentioning

confidence: 99%

Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms

et al. 2023

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“…The remaining instance can be solved by an exact algorithm in polynomial time. Hauptmann [16] showed that the same scheme actually yields even an EPTAS for the ǫ-dense Steiner Tree problem.…”

Section: The Subdense Steiner Tree Problemmentioning

confidence: 96%

“…They showed that for every ǫ > 0 the ǫ-dense Steiner Tree problem admits a PTAS. Later on, Hauptmann [16] showed that the same scheme even yields an efficient PTAS for the ǫ-dense Steiner Tree problem.…”

Section: Set Covermentioning

confidence: 99%

Approximating Subdense Instances of Covering Problems

Cardinal

Karpiński

Schmied

et al. 2011

Electronic Notes in Discrete Mathematics

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We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Ω(n/ψ(n)) for some function ψ(n) = ω(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest.

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“…The number of papers on approximation of some variant of Steiner type problem is enormous and rapidly growing. Therefore we refer the reader to the online compendium [25] for the current approximation state of various Steiner type problems.…”

Section: Related Workmentioning

confidence: 99%

Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

Suchý

2016

Algorithmica

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In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T | + k, the total number of vertices in the constructed subgraph.In this paper we present several polynomial time applicable reduction rules for Planar Steiner Tree. In an instance reduced with respect to the presented reduction rules, the number of terminals |T | is at most quadratic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.

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On the Approximability of Dense Steiner Tree Problems

Cited by 4 publications

References 0 publications

Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms

Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms

Approximating Subdense Instances of Covering Problems

Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

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