A. Zelikovsky, A series of approximation algorithms for the acyclic directed Steiner tree problem, Algorithmica, 18: 99-110, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....well, but they had no good worst case approximation guarantees. There are several applications of these problems in networks and VLSI design. This problem is known to be NP hard [4] There are many papers that give approximation algorithms for this problem starting with the work by Zelikovsky [7], and more recently by Charikar, Chekuri, Cheung, Dai, Current address: ITG Inc. 44 Farnsworth Str. Boston, MA 02210 Part of this work was done while visiting the University of Maryland. E mail:lzosin itginc.com. Computer Science Dept. University of Maryland, College Park, MD 20742. ....

....CCR 0113192. Goel, Guha and Li [2] In these papers a polynomial time algorithm is developed with an approximation ratio of O(k ) for any fixed ffl 0. In [2] this is extended to a poly log approximation algorithm, but the running time is not polynomial. For other results, see the references [2, 7]. This problem is a generalization of the group Steiner tree problem, for which a polylog approximation was obtained by Garg, Konjevod and Ravi [5] In the group Steiner tree problem, we are given an undirected graph G with subsets S j V (j = 1 : k) and we wish to compute a minimum weight ....

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A. Zelikovsky, "A series of approximation algorithms for the acyclic directed Steiner tree problem", Algorithmica, 18(1):99-110, (1997).

....is likely to total be NP hard. 12] provide an exhaustive algorithm which runs in exponential time. Since the algorithm is impractical for real life applications, they also describe an approximation algorithm. They reduce the problem to a directed Steiner tree problem and apply the algorithm of [25]. The solution is O(1(1 E) times worse than the optimal, where 0 1. The second operator is the shared scan index based join. Let q, q2 ( and let v be a materialized view which can answer both queries. Assume that each dimension table has bitmap join indices that map the join attributes to ....

Zelikovsky A., "A series of approximation algorithms for the acyclic directed Steiner tree problem", Algorithmica 18, 1997.

....a factor better than ln k, in which k is the number of terminals [4] On the other hand, it is trivial to obtain an approximation factor of k by connecting all terminals to the root with a shortest path. For the special case of directed acyclic graphs, as in the ow scheduling problem, Zelikovsky [24] has proposed an approximation algorithm which achieves a factor of k 1=i (2 ln k) i 1 for any integer i 0. The runtime of that algorithm is O( k i n i 1 ) where k denotes the number of terminals, and n is the number of nodes in the graph. The stands for the complexity of the ....

....two, then this tree is optimal. If there are more terminals, then we construct this tree and assign the path to the closest terminal. Then we repeat the process with the remaining terminals. The process is shown in procedure 5. This algorithm is simpler than the algorithms proposed by Zelikovsky [24] and Charikar [2] Its complexity is O(km k 2 n) instead of the O(n 3 log k ) time required for the O(log 2 k) approach by Charikar. The performance is guaranteed to be within a factor p k of the optimal because it 6 The factor m comes from the time required to calculate the shortest ....

A. Zelikovsky. A Series of Approximation Algorithms for the Acyclic Directed Steiner Tree Problem. Algorithmica, 18:99-110, 1997. 46

....to be NP hard. LOY00] provide an exhaustive algorithm which runs in exponential time. Since the algorithm is impractical for real life applications, they also describe an approximation algorithm. They reduce the problem to a directed Steiner tree problem and apply the algorithm of Zelikovsky [Zeli97]. The solution is O( Q # ) times worse than the optimal, where 0 # # 1. The second operator is the shared scan index based join. Similar to the previous case, a set of queries are answered by the same view v, but there are bitmap indexes that can be used to accelerate all queries. The read ....

Zelikovsky A., "A series of approximation algorithms for the acyclic directed Steiner tree problem", Algorithmica 18, 1997.

....k where k is the number of terminals unless NP DT IME[n O(log log n) 9] It is also easy to obtain an approximation factor of k, by connecting every terminal to the root via a shortest path. The only known polynomial time approximation algorithm even for a special case is due to Zelikovsky [27], where he gives an approximation algorithm which achieves a ratio of (2 ln k) i Gamma1 k 1=i for any i 0 for directed acyclic graphs. Zelikovsky further conjectures that no subpolynomial approximation guarantees are possible unless P = NP . There is a simple reduction from arbitrary ....

....graphs, therefore Zelikovsky s results carry over to the general case. In this paper, we present a factor i(i Gamma 1)k 1=i approximation algorithm which runs in time O(n i k 2i ) for any i 0 for the directed Steiner tree problem. Our approach is simpler compared to that of Zelikovsky [27]. Further, setting i = log k, we obtain an O(log 2 k) approximation in O(n 3 log k ) time. Our result gives evidence that the problem can be approximated to within polylogarithmic factors in polynomial time contrary to Zelikovsky s conjecture. This is our main contribution. Many of the ....

[Article contains additional citation context not shown here]

A. Zelikovsky, "A series of approximation algorithms for the Acyclic Directed Steiner Tree problem", Algorithmica, 18: 99-110 (1997). 15

....ln k where k is the number of terminals unless NP DT IME[n O(loglogn) 8] It is also easy to obtain an approximation factor of k, by connecting every terminal to the root via a shortest path. The only known polynomial time approximation algorithm even for a special case is due to Zelikovsky [23], where he gives an approximation algorithm which achieves a ratio of (2 ln k) i Gamma1 k 1=i for any i 0 for directed acyclic graphs. No 1 algorithms were known for the case of general directed graphs. Zelikovsky further conjectures that no subpolynomial approximation guarantees are ....

....conjectures that no subpolynomial approximation guarantees are possible unless P = NP . In this paper, we present a factor i(i Gamma 1)k 1=i approximation algorithm for any i 0 for the Directed Steiner tree problem. Not only is our approach new and simpler compared to that of Zelikovsky [23], but also generalizes to arbitrary directed graphs. Further, setting i = log k, we obtain an O(log 2 k) approximation in O(n 3 log k ) time. This fact gives strong evidence that the problem can be approximated to within poly logarithmic factors in polynomial time contrary to Zelikovsky s ....

[Article contains additional citation context not shown here]

A. Zelikovsky, "A series of approximation algorithms for the Acyclic Directed Steiner Tree problem", Algorithmica, 18: 99-110 (1997). ii

....k where k is the number of terminals unless NP DT IME[n O(log log n) 9] It is also easy to obtain an approximation factor of k, by connecting every terminal to the root via a shortest path. The only known polynomial time approximation algorithm even for a special case is due to Zelikovsky [27], where he gives an approximation algorithm which achieves a ratio of (2 ln k) i 1 k 1=i for any i 0 for directed acyclic graphs. Zelikovsky further conjectures that no subpolynomial approximation guarantees are possible unless P = NP . There is a simple reduction from arbitrary directed ....

....to acyclic graphs, therefore Zelikovsky s results carry over to the general case. In this paper, we present a factor i(i 1)k 1=i approximation algorithm which runs in time O(n i k 2i ) for any i 0 for the directed Steiner tree problem. Our approach is simpler compared to that of Zelikovsky [27]. Further, setting i = log k, we obtain an O(log 2 k) approximation in O(n 3 log k ) time. Our result gives evidence that the problem can be approximated to within polylogarithmic factors in polynomial time contrary to Zelikovsky s conjecture. This is our main contribution. Many of the ....

[Article contains additional citation context not shown here]

A. Zelikovsky, \A series of approximation algorithms for the Acyclic Directed Steiner Tree problem", Algorithmica, 18: 99-110 (1997). 15

....k where k is the number of terminals unless NP DT IME[n O(log log n) 8] It is also easy to obtain an approximation factor of k, by connecting every terminal to the root via a shortest path. The only known polynomial time approximation algorithm even for a special case is due to Zelikovsky [25], where he gives an approximation algorithm which achieves a ratio of (2 ln k) i Gamma1 k 1=i for any i 0 for directed acyclic graphs. Zelikovsky further conjectures that no subpolynomial approximation guarantees are possible unless P = NP . In this paper, we present a factor i(i Gamma ....

....approximation guarantees are possible unless P = NP . In this paper, we present a factor i(i Gamma 1)k 1=i approximation algorithm which runs in time O(n i k 2i ) for any i 0 for the directed Steiner tree problem. Not only is our approach simpler compared to that of Zelikovsky [25], but also generalizes to arbitrary directed graphs. Further, setting i = log k, we obtain an O(log 2 k) approximation in O(n 3 log k ) time. Our result gives strong evidence that the problem can be approximated to within poly logarithmic factors in polynomial time contrary to Zelikovsky s ....

[Article contains additional citation context not shown here]

A. Zelikovsky, "A series of approximation algorithms for the Acyclic Directed Steiner Tree problem", Algorithmica, 18: 99-110 (1997).

....it in each group) can be computed in time less than . 7 If a node v is reachable from a node u in G, then in the transitive closure of G there is an arc (u; v) whose cost is equal to the cost of the minimum directed u to v path. 8 Although the case of acyclic directed graphs was considered in [24], the technique suggested there does not generalize to arbitrary directed graphs. 19 The time complexity of the Partial d Star Heuristic (Figure 10) is O(jV j d Gamma1 Delta k d ) where k is the number of groups, and d is the depth. Therefore, the Rooted d Star Heuristic (Figure 6) has ....

A. Z. Zelikovsky. A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica, 18:99--110, 1997. 27

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A. Zelikovsky, A series of approximation algorithms for the acyclic directed Steiner tree problem, Algorithmica, 18: 99-110, 1997.

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