## A series of approximation algorithms for the Acyclic Directed Steiner Tree problem (1997)

Venue: | Algorithmica |

Citations: | 38 - 1 self |

### BibTeX

@ARTICLE{Zelikovsky97aseries,

author = {Alexander Zelikovsky},

title = {A series of approximation algorithms for the Acyclic Directed Steiner Tree problem},

journal = {Algorithmica},

year = {1997},

volume = {18},

pages = {99--110}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract Given an acyclic directed network, a subset S of nodes (terminals), and a root r, the acyclic directed Steiner tree problem requires a minimum-cost subnetwork which contains paths from r to each terminal. It is known that unless NP ` DT IME[npolylogn] no polynomial-time algorithm can guarantee better then (ln k)=4- approximation, where k is the number of terminals. In this paper we give an O(kffl)-approximation algorithm for any ffl? 0. This result improves the previously known k-approximation.

### Citations

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Citation Context ...l-time algorithm for ADSP based on embedding of a graph in a d-dimensional rectilinear metric was given in [11]. Most of cases of the general Steiner tree problem (NSP, NWSP, ADSP, DSP) are N P -hard =-=[7]-=-, so many approximation algorithms have appeared in the last two decades. The quality of an approximation algorithm is measured by its performance ratio: an upper bound on the ratio between the achiev... |

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Citation Context ...? 1, \Lambda Institute of Mathematics, Akademiei 5, Kishinev, 277028, Moldova, email: 17azz@mathem.moldova.su. Research partially supported by Volkswagen-Stiftung 1sffl-approximation is N P -complete =-=[1]-=-. For NWSP, a 2 ln k-approximation algorithm was designed [8]. From the other side, the famous set cover problem may be embedded in NWSP. This implies that NWSP cannot be approximated to within less t... |

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Citation Context ...signed [8]. From the other side, the famous set cover problem may be embedded in NWSP. This implies that NWSP cannot be approximated to within less than 14 ln k-factor unless DT IM E[npolylogn] ' N P =-=[9]-=-. Therefore, the only question for these problems is still open: what exact constants separate polynomially solvable and N P -complete approximations? For NSP, this constant is at most 1 + ln 2 ss 1:6... |

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Citation Context ...er tree and its cost, respectively. The elements of the set S are called terminals. The number of terminals is denoted by k. ADSP is also known as the Steiner arborescence problem in acyclic networks =-=[6]-=-. It has various practical applications. The most important occurs in biology while constructing phylogenetic trees [4]. A number of papers are devoted to the case of a digraph embedded in a d-dimensi... |

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Citation Context ...been made in the last few years [16]. The approximation complexity of NSP and NWSP has already been determined. NSP belongs to M AXSN P -class [3], so a constant factor approximation algorithm exists =-=[14]-=- and for some ffl ? 1, \Lambda Institute of Mathematics, Akademiei 5, Kishinev, 277028, Moldova, email: 17azz@mathem.moldova.su. Research partially supported by Volkswagen-Stiftung 1sffl-approximation... |

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Citation Context ...f all paths and f(T ) = d(T ). The Rayward-Smith's heuristic (RSH) [13]. K contains all stars and f(T ) = d(T)r\Gamma 1 , where r is the number of leaves of T . The generalized greedy heuristic (GGH) =-=[17]-=-. K consists of trees with 3 terminals and f(T ) = d(T ) \Gammas(M0(S) \GammasM0(S=T )). The size-restricted relative greedy heuristic (SRGH) [18]. K = Kr contains all trees with at most r terminals. ... |

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Citation Context ... 277028, Moldova, email: 17azz@mathem.moldova.su. Research partially supported by Volkswagen-Stiftung 1sffl-approximation is N P -complete [1]. For NWSP, a 2 ln k-approximation algorithm was designed =-=[8]-=-. From the other side, the famous set cover problem may be embedded in NWSP. This implies that NWSP cannot be approximated to within less than 14 ln k-factor unless DT IM E[npolylogn] ' N P [9]. There... |

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Citation Context ...algorithms Al. For brevity, the sets and its cardinalities will have the same denotations. Since the graph G is acyclic, we can apply the O(E + V log V )-procedure of finding M ST (S) due to Mehlhorn =-=[10]-=-. Thus, A1 runs in time O(E + V log V ). For A2, we need to know all distances between S and V \GammasS. This can be done in time O(E + V log V ) by adding an auxiliary node with zero cost edges to S ... |

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Citation Context ...by k. ADSP is also known as the Steiner arborescence problem in acyclic networks [6]. It has various practical applications. The most important occurs in biology while constructing phylogenetic trees =-=[4]-=-. A number of papers are devoted to the case of a digraph embedded in a d-dimensional rectilinear metric. For d = 2, a fast and effective heuristic was proposed in [12]; however, this case has not yet... |

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Citation Context ... constructing phylogenetic trees [4]. A number of papers are devoted to the case of a digraph embedded in a d-dimensional rectilinear metric. For d = 2, a fast and effective heuristic was proposed in =-=[12]-=-; however, this case has not yet been shown to be N P -hard. An exact exponential-time algorithm for ADSP based on embedding of a graph in a d-dimensional rectilinear metric was given in [11]. Most of... |

59 |
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Citation Context ...s for approximation complexity of N P -hard problems has been made in the last few years [16]. The approximation complexity of NSP and NWSP has already been determined. NSP belongs to M AXSN P -class =-=[3]-=-, so a constant factor approximation algorithm exists [14] and for some ffl ? 1, \Lambda Institute of Mathematics, Akademiei 5, Kishinev, 277028, Moldova, email: 17azz@mathem.moldova.su. Research part... |

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Citation Context ...sidering different definitions of a class K and a criterion function f. The minimum spanning tree heuristic (MSTH) [14]. K consists of all paths and f(T ) = d(T ). The Rayward-Smith's heuristic (RSH) =-=[13]-=-. K contains all stars and f(T ) = d(T)r\Gamma 1 , where r is the number of leaves of T . The generalized greedy heuristic (GGH) [17]. K consists of trees with 3 terminals and f(T ) = d(T ) \Gammas(M0... |

28 |
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Citation Context ... a2 = 1 for NWSP, ADSP. RSH gives a1 ^ 5=3 and a1 \Deltasa2 ^ 2 for NSP [15] and a1 \Deltasa2 ^ 2 log k for NWSP [8]. GGH gives a1 ^ 5=3 and a1 \Deltasa2 ^ 11=6 for NSP [17]. SRGH gives limr!1 a1 = 1 =-=[5]-=- and limr!1 a2 = 1 + ln 2 for NSP [18]. In other words, it induces a series of approximation algorithms for NSP with the limit performance ratio (1 + ln 2). In this paper we present a level-restricted... |

16 | Approaching the 5/4-approximation for Rectilinear Steiner Trees
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Citation Context ...ete approximations? For NSP, this constant is at most 1 + ln 2 ss 1:69 [18]. For the euclidean and rectilinear subcases of NSP, these constants are at most 1 + ln 2p3 ss 1:1438 [18] and 6148 ss 1:271 =-=[2]-=-, respectively. The approximation complexity of ADSP and DSP is still unknown. The only thing we can say that the set cover problem can be transformed to ADSP, so these problems are not easier to appr... |

6 |
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Citation Context ...posed in [12]; however, this case has not yet been shown to be N P -hard. An exact exponential-time algorithm for ADSP based on embedding of a graph in a d-dimensional rectilinear metric was given in =-=[11]-=-. Most of cases of the general Steiner tree problem (NSP, NWSP, ADSP, DSP) are N P -hard [7], so many approximation algorithms have appeared in the last two decades. The quality of an approximation al... |

4 |
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Citation Context ...n bounds for the ratios a1 and a2 for the above heuristics embedded in GCF. MSTH gives a1 ^ 2 and a2 = 1 for NSP, and a1 ^ k and a2 = 1 for NWSP, ADSP. RSH gives a1 ^ 5=3 and a1 \Deltasa2 ^ 2 for NSP =-=[15]-=- and a1 \Deltasa2 ^ 2 log k for NWSP [8]. GGH gives a1 ^ 5=3 and a1 \Deltasa2 ^ 11=6 for NSP [17]. SRGH gives limr!1 a1 = 1 [5] and limr!1 a2 = 1 + ln 2 for NSP [18]. In other words, it induces a seri... |

4 |
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(Show Context)
Citation Context ...Therefore, the only question for these problems is still open: what exact constants separate polynomially solvable and N P -complete approximations? For NSP, this constant is at most 1 + ln 2 ss 1:69 =-=[18]-=-. For the euclidean and rectilinear subcases of NSP, these constants are at most 1 + ln 2p3 ss 1:1438 [18] and 6148 ss 1:271 [2], respectively. The approximation complexity of ADSP and DSP is still un... |

2 |
Recent developements on the approximability of combinatorial problems
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(Show Context)
Citation Context ...rformance ratio may depend on the number of terminals. From the other side, significant progress in lower bounds for approximation complexity of N P -hard problems has been made in the last few years =-=[16]-=-. The approximation complexity of NSP and NWSP has already been determined. NSP belongs to M AXSN P -class [3], so a constant factor approximation algorithm exists [14] and for some ffl ? 1, \Lambda I... |