Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Dvořák, Pavel; Feldmann, Andreas Emil; Knop, Dušan; Masařík, Tomáš; Toufar, Tomáš; Veselý, Pavel

doi:10.4230/lipics.stacs.2018.26

Cited by 6 publications

(17 citation statements)

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“…Despite our former believes that were supported by the results of Dvořák et al [12], the answer for the third question seems to be also negative. Figure 1 indicate that there is no threshold point for neither classical nor improved stars.…”

Section: Our Contributioncontrasting

confidence: 58%

“…A recent result of Dvořák et al [12], which proposes a novel algorithm in the framework of parameterized approximations, also falls in the framework suggested in Algorithm 1. Surprisingly, their algorithm uses an unbounded value of the parameter k, the terminal-size of a best star.…”

Section: Unbounded Size Of a Best Starmentioning

confidence: 99%

“…In order to answer our questions we have implemented the algorithm for Steiner Tree of Dvořák et al [12] (with some modifications in order to be able to run some experiments). It is worth noting that even though our work is experimental the core of our work is mostly proof-of-concept, that is, we experimentally investigate and evaluate quantitative features of star contractions.…”

Section: Our Contributionmentioning

confidence: 99%

“…In this section we describe our implementation and improvements of finding the Best Star as proposed in [12] (Section 2.2), the approximate algorithms used to finish the solution after the application of the Best Star Algorithm (Section 2.3), and also the heuristics we used to preprocess the instances (Section 2.1). Last but not least, we discuss a few simple yet in practice well-performing modification of Zelikovsky's algorithm (Section 2.3).…”

Section: Implementation Details and Heuristicsmentioning

confidence: 99%

“…As already mentioned in Section 1, Algorithm 1 repeatedly finds and contracts a so called best star (according to a certain ratio) in G. Here, we first give the definition used by Dvořák et al [12]. Later, we discuss a further additional practical extension of the former definition and describe our implementation in detail.…”

Section: Finding the Best Starmentioning

confidence: 99%

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Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

Hušek,

Knop,

Masařík

2020

Preprint

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In the Steiner Tree problem we are given an edge weighted undirected graph G = (V, E) and a set of terminals R ⊆ V . The task is to find a connected subgraph of G containing R and minimizing the sum of weights of its edges. Steiner Tree is well known to be NP-complete and is undoubtedly one of the most studied problems in (applied) computer science. Steiner Tree has been analyzed within many theoretical frameworks as well as from the practical perspective.We observe that many approximation algorithms for Steiner Tree 1. follow a similar scheme (meta-algorithm) and 2. perform (exhaustively) a similar routine which we call star contraction.Here, by a star contraction, we mean finding a star-like subgraph in (the metric closure of) the input graph minimizing the ratio of its weight to the number of contained terminals minus one. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. Zelikovsky's approximation algorithm follows a similar workflow, finding the best star among those containing three terminals.We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction. Our experiments suggest the following:if the algorithm performs star contractions exhaustively, the quality of the solution is usually slightly better than Zelikovsky's 11/6-approximation algorithm, on average the quality of the solution returned by the MST-approximation algorithm improves with every star contraction, the same holds for iterated MST (MST+), which outperforms MST in every measurement while keeping very fast running time (on average ∼ 3× slower than MST), on average the quality of the solution obtained by exhaustively performing star contraction is about 16% better than the initial MST-approximation, and we propose a more precise way to find the so-called improved stars which yield a slightly better solution within a comparable running time (on average ∼ 3× slower). Furthermore, we propose two improvements of Zelikovsky's 11/6-approximation algorithm and we empirically confirm that the quality of the solution returned by any of these is better than the one returned by the former algorithm. However, such an improvement is exchanged for a slower running time (up to a multiplicative factor of the number of terminals).

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Section: Our Contributioncontrasting

confidence: 58%

Section: Unbounded Size Of a Best Starmentioning

confidence: 99%

Section: Our Contributionmentioning

confidence: 99%

Section: Implementation Details and Heuristicsmentioning

confidence: 99%

Section: Finding the Best Starmentioning

confidence: 99%

See 3 more Smart Citations

Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

Hušek,

Knop,

Masařík

2020

Preprint

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Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Chitnis¹,

Feldmann

Hajiaghayi

et al. 2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a vertex-weighted directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . .t k } of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆ V of minimum weight such that G[H] contains a t i → t j path for each i = j. The problem is NP-hard, but Feldman and Ruhl (FOCS '99; SICOMP '06) gave a novel n O(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs.in the exponent over the algorithm of Feldman and Ruhl.• Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f (k) • n o( √ k) algorithm for any computable function f , unless the Exponential Time Hypothesis (ETH) fails.

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FPT-approximation for FPT Problems

Lokshtanov¹,

Misra²,

Ramanujan³

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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Over the past decade, many results have focused on the design of parameterized approximation algorithms for W[1]-hard problems. However, there are fundamental problems within the class FPT for which the best known algorithms have seen no progress over the course of the decade; some of them have even been proved not to admit algorithms that run in time 2 O(k) n O(1) under the Exponential Time Hypothesis (ETH) or (c − ) k n O(1) under the Strong ETH (SETH). In this paper, we expand the study of FPT-approximation and initiate a systematic study of FPT-approximation for problems that are FPT. We design FPT-approximation algorithms for problems that are FPT, with running times that are significantly faster than the corresponding best known FPT-algorithm, and while achieving approximation ratios that are significantly better than what is possible in polynomial time.• We present a general scheme to design 2 O(k) n O(1) -time 2-approximation algorithms for cut problems.In particular, we exemplify it for Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut.• Further, we extend our scheme to obtain FPT-time O(1)-approximation algorithms for weighted cut problems, where the objective is to obtain a solution of size at most k and of minimum weight. Here, we present two approaches. The first approach achieves 2 O(k) n O(1) -time constant-factor approximation, which we exemplify for all problems mentioned in the first bullet. The other leads to an FPT-approximation Scheme (FPT-AS) for Weighted Directed Feedback Vertex Set.• Additionally, we present a combinatorial lemma that yields a partition of the vertex set of a graph to roughly equal sized sets so that the removal of each set reduces its treewidth substantially, which may be of independent interest. For several graph problems, use this lemma to design c w n O(1) -time (1 + )approximation algorithms that are faster than known SETH lower bounds, where w is the treewidth of the input graph. Examples of such problems include Vertex Cover, Component Order Connectivity, Bounded-Degree Vertex Deletion and F-Packing for any family F of bounded sized graphs.• Lastly, we present a general reduction of problems parameterized by treewidth to their versions parameterized by solution size. Combined with our first scheme, we exemplify it to obtain c w n O(1) -time bi-criteria approximation algorithms for all problems mentioned in the first bullet.

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Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Cited by 6 publications

References 0 publications

Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

FPT-approximation for FPT Problems

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