A threshold of ln <i>n</i> for approximating set cover (preliminary version)

Feige, Uriel

doi:10.1145/237814.237977

Cited by 164 publications

(18 citation statements)

References 13 publications

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“…Let OPT(ϕ) denote the maximum number of clauses of ϕ satisfied by a truth assignment. It is known that it is NPhard to distinguish between instances ϕ of MAX-E3SAT(5) for which OPT(ϕ) = m and instances ϕ for which OPT(ϕ) < (1 − )m for some fixed constant 0 < < 1 [13].…”

Section: Hardness Of Approximating Smftmentioning

confidence: 99%

Hardness and Approximation of the Survivable Multi-Level Fat Tree Problem

Ngo

Nguyen

2009

Ieee Infocom 2009

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Abstract-With the explosive deployment of "triple play" (voice, video and data services) over the same access network, guaranteeing a certain-level of survivability for the access network is becoming critical for service providers. The problem of economically provisioning survivable access networks has given rise to a new class of network design problems, including the so-called SURVIVABLE MULTI-LEVEL FAT TREE problem (SMFT).We show that two special cases of SMFT are polynomialtime solvable, and present two approximation algorithms for the general case. The first is a combinatorial algorithm with approximation ratio min{ L/2 + 1, 2 log 2 n} where L is the longest Steiner path length between two terminals, and n is the number of nodes. The second is a primal-dual (2∆s + 2)-approximation algorithm where ∆s is the maximum Steiner degree of terminals in the access network. We then show that approximating SMFT to within a certain constant c > 1 is NPhard, even when all edge-weights of G are 1, L ≤ 10, and ∆s ≤ 3. Finally, we experimentally show that the approximation algorithms perform extremely well on random instances of the problem.

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Section: Hardness Of Approximating Smftmentioning

confidence: 99%

Hardness and Approximation of the Survivable Multi-Level Fat Tree Problem

Ngo

Nguyen

2009

Ieee Infocom 2009

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“…In [1] it was proved that for each r > 0, the Set Covering problem cannot be approximated within factor of (1 -e) In N in polynomial time unless NP C DTIME(n ~176 log2 n)).…”

Section: Np C Dtime(np~176mentioning

confidence: 99%

“…Let T E T. Decision Tree Constructing problem is the problem of searching decision tree with minimal depth for the table T. Using mentioned result from [1] it is not difficult to prove that for each e > 0, the Decision Tree Constructing problem cannot be approximated within factor of (1 -c) In R(T) in polynomial time unless NP C DTIME(n ~176 log2 n)). Taking into account that algorithms UR,h, UG,h, UH,h have polynomial time complexity, using inequalities G(T) < H(T) < R(T) (which are true for each T E T) and using Theorem 6 we obtain that unless NP C DTIME(n ~176176 then algorithms UR,h, UG,h, UH, h are close to unimprovable approximate polynomial algorithms for Decision Tree Constructing problem solving.…”

Section: Np C Dtime(np~176mentioning

confidence: 99%

“…In these algorithms different measures for time complexity of decision trees and different measures for uncertainty of decision tables are used. New results about precision of polynomial approximate algorithms for covering problem solving [1,2] show that some of considered algorithms for decision tree constructing are, apparently, close to unimprovable. …”

mentioning

confidence: 99%

“…Obtained results were published only in Russian. New bounds on precision of polynomial approximate algorithms for covering problem solving [1,2] show that some of considered in [4] algorithms are, apparently, close to unimprovable polynomial approximate algorithms for constructing decision trees with minimal depth. This paper contains a survey of some results from [3, 4] (definitions of complexity and uncertainty measures, description of algorithms for decision tree constructing, bounds on precision of these algorithms), some results from [5] (bounds on complexity of considered algorithms and bounds on precision of algorithms for one important class of decision tables), and reasons about closeness of some algorithms to unimprovable.…”

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confidence: 99%

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Algorithms for constructing of decision trees

Moshkov

1997

Principles of Data Mining and Knowledge Discovery

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Abstract. Decision trees are widely used in different applications for problem solving and for knowledge representation. In the paper algorithms for decision tree constructing with bounds on complexity and precision are considered. In these algorithms different measures for time complexity of decision trees and different measures for uncertainty of decision tables are used. New results about precision of polynomial approximate algorithms for covering problem solving [1,2] show that some of considered algorithms for decision tree constructing are, apparently, close to unimprovable. IntroductionIn 1983 the paper [4] was published (short variant was published in 1982 [3]) which contained bounds on time complexity of decision trees and algorithms for decision tree constructing (in this paper decision trees were named conditional tests). In algorithms different measures for time complexity of decision trees (depth, weighted depth and others) and different measures for uncertainty of decision tables were used. Bounds on precision for these algorithms were considered. These algorithms resemble on algorithms of J.R. Quinlan [10, 11] however they were proposed independently. Obtained results were published only in Russian. New bounds on precision of polynomial approximate algorithms for covering problem solving [1,2] show that some of considered in [4] algorithms are, apparently, close to unimprovable polynomial approximate algorithms for constructing decision trees with minimal depth. This paper contains a survey of some results from [3, 4] (definitions of complexity and uncertainty measures, description of algorithms for decision tree constructing, bounds on precision of these algorithms), some results from [5] (bounds on complexity of considered algorithms and bounds on precision of algorithms for one important class of decision tables), and reasons about closeness of some algorithms to unimprovable.Mentioned results are useful for obtaining bounds on time complexity of decision trees [7,8]. Also these results may be useful in data mining and knowledge discovery for decision tree constructing. The considered algorithms allow variations of complexity and uncertainty measures in broad bounds: the statements

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Improving Minimum Cost Spanning Trees by Upgrading Nodes

Krumke

Marathe

Noltemeier

et al. 1999

Journal of Algorithms

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We study budget constrained network upgrading problems. We are given an Ž . undirected edge-weighted graph G s V, E , where node¨g V can be upgraded Ž . at a cost of c¨. This upgrade reduces the weight of each edge incident on¨. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a minimum spanning tree of weight no more than a given budget D. The results obtained in the paper include 1. On the positive side, we provide a polynomial time approximation algorithm for the above upgrading problem when the difference between the maximum and minimum edge weights is bounded by a polynomial in n, the number of nodes in the graph. The solution produced by the algorithm satisfies the budget con-Ž . straint, and the cost of the upgrading set produced by the algorithm is O O log n times the minimum upgrading cost needed to obtain a spanning tree of weight at most D.In contrast, we show that, unless NP : DTIME n , there can be no polynomial time approximation algorithm for the problem that produces a solution with upgrading cost at most ␣ -ln n times the optimal upgrading cost Ž . even if the budget can be violated by a factor f n , for any polynomial time Ž . Ž .k computable function f n . This result continues to hold, with f n s n being any polynomial, even when the difference between the maximum and minimum edge weights is bounded by a polynomial in n.3. Finally, we show that using a sample binary search over the set of admissible values, the dual problem can be solved with an appropriate performance guarantee.

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A threshold of ln n for approximating set cover (preliminary version)

Cited by 164 publications

References 13 publications

Hardness and Approximation of the Survivable Multi-Level Fat Tree Problem

Hardness and Approximation of the Survivable Multi-Level Fat Tree Problem

Algorithms for constructing of decision trees

Improving Minimum Cost Spanning Trees by Upgrading Nodes

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