. B. Berger, J. Rompel, P. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Proc. 1989 IEEE FOCS, 54-59.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the primal dual schema, to obtain simple parallel approximation algorithms for the set cover problem and its generalizations. Our algorithms use randomization, and our randomized voting lemmas may be of independent interest. Fast parallel approximation algorithms were known before for set cover [BRS89] [LN93] though not for all its generalizations. Work done while the author was a graduate student at the University of California, Berkeley, supported by NSF PYI Award CCR 88 96202and NSF grant IRI 91 20074. Part of this work was done when visiting IIT, Delhi. y Partial support provided by ....

....ratio of O(logn) for this problem has been shown to be essentially tight by Lund and Yannakakis [LY92] 2 More recently, Feige [Fe95] has shown that approximation ratios better than ln n are unlikely. The first parallel algorithm for approximating set cover is due to Berger, Rompel and Shor [BRS89], who found an RNC 5 algorithm with an approximation guarantee of O(logn) Further, this algorithm can be derandomized to obtain an NC 7 algorithm with the same approximation guarantee. Luby and Nisan [LN93] building on the work of [PST91] have obtained a (1 ffl) factor (for any constant ffl ....

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Berger, B., Rompel, J., and Shor, P. Efficient NC Algorithms for Set Cover with applications to Learning and Geometry. 30 th IEEE Symposium on the Foundations of Computer Science (1989), proceedings pp 54-59.

....dual feasible solutions to the problem, and based on this we describe a fast parallel approximation algorithm. We use ideas that were previously employed in similar contexts by [1, Berger, Rompel, Shor] 8, Plotkin, Shmoys, Tardos] and [2, Chazelle, Friedman] We use the general idea also used in [1] of incrementing the values of many variables in parallel, and we use the general idea also used in [8] and [2] of changing the weight function on the constraints by an amount exponential in the change in the variables. The overall method we introduce is novel in several respects, including the ....

.... feasible solution such that sum(opt(z) min z is primal feasible fsum(z)g: We consider also the dual of such problems: The Dual Problem : The objective is to find q = hq 1 ; q m i that maximizes sum(q) X j b j Delta q j Our results are a slight improvement over those in [1] in the sense that our multiplicative constant is 1 ffl, where ffl is an input parameter, whereas their multiplicative constant is fixed to something like 2. 2 subject to the following constraints: ffl For all j, q j 0. ffl For all i, P j c i;j Delta q j d i . We say q is dual feasible ....

Berger, B., Rompel, J., Shor, P., "Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry", 30 th Annual Symposium on Foundations of Computer Science, pp. 54-59, 1989.

....dual feasible solutions to the problem, and based on this we describe a fast parallel approximation algorithm. We use ideas that were previously employed in similar contexts by [1, Berger, Rompel, Shor] 5, Plotkin, Shmoys, Tardos] and [2, Chazelle, Friedman] We use the general idea also used in [1] of incrementing the values of many variables in parallel, and we use the general idea also used in [5] and [2] of changing the weight function on the constraints by an amount exponential in the change in the variables. The overall method we introduce is novel in several respects, including the ....

.... log( Delta) This is essentially optimal (up to NPcompleteness [4, Lund, Yannakakis] and matches results of [1, Berger, Rompel, Shor] 1 In this case finding the set cover itself is also possible using e.g. ideas found in [6, Raghavan] 1 Our results are a slight improvement over those in [1] in the sense that our multiplicative constant is 1 ffl, where ffl is an input parameter, whereas their multiplicative constant is fixed to something like 2. 2 2 The problem Throughout this paper, n is the number of variables and m is the number of constraints (not including constraints of ....

Berger, B., Rompel, J., Shor, P., "Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry", 30 th Annual Symposium on Foundations of Computer Science, pp. 54-59, 1989.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., volume 30, pages 54--59, 1989.

....Megiddo [KM94b] and further developed by Karger and Koller [KK94] Our bibliography is by no means complete here; the reader may consult some of the recent papers mentioned here for references to other important works in this area. The k wise independence methods were applied by Berger et al. BRS94] and later by Goodrich [Goo93] Goo96] by Amato et al. AGR95] and by Mahajan et al. MRS97] for parallelization of derandomized geometric algorithms. Mulmuley [Mul96] extends earlier work of Karloff and Raghavan [KR93] on limiting random resources using bounded independence distributions and ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. J. Comput. Syst. Sciences, 49:454--477, 1994.

....of Lund and Yannakakis [9] shows that this problem cannot be approximated in P with ratio c log 2 n for any c 1=4 unless NP = DTIME(n O(1) However there is known a polynomial time algorithm that finds a logarithmic factor approximation. The following lemma has been shown by Berger et al. [3]. Lemma 17. For any 0, there is an NC algorithm for the weighted set cover problem that runs in O(log 4 n log m log 2 (nm) 6 ) time, uses O(n P m i=1 j Y i j) processors, and produces a cover of weight at most (1 ) log n times the weight of a minimum cover. 5.2 The NC Algorithm ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. 30th FOCS, pp. 54--59, 1989. The full version is to appear in J. Algorithms.

....she conjectures that there is no polynomial time c approximation algorithm for any c 2 unless P=NP. Chv atal s weighted set cover algorithm guarantees a set cover of weight at most ln Delta times the minimum, where Delta is the maximum set size [Ch79, Lo75, Jo74] Berger, Rompel, and Shor [BRS89] give a parallel algorithm that guarantees a factor of (1 ffl) ln Delta. Their algorithm uses a linear number of processors and runs in polylogarithmic time with some restrictions on the weights. The intuition behind our complexity analysis relies on a lemma of general interest for parallel ....

B. Berger, J. Rompel, and P. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. In Proc. 30th Annual Symp. on Foundations of Computer Science, pages 54--59, October 1989. Research Triangle Park, NC. To appear in Journal of Algorithms (Special issue on FOCS-89).

....of the whole algorithm) and the size of the resulting (1=r) approximation. Both these sizes will be roughly (r pa log r) 1= 1=20fl) so the exponent converges to 2 as fl 0. The (1=r) net is again computed from a (1=2r) approximation. This time we can use e.g. the result of Berger et al. [BRS89] on parallelizing the set covering problem, from which our claim follows. 2 Remark. The most important special case of the above theorem is for a fixed r. In such a situation, we can produce an NC algorithm with a linear number of processors without the machinery of [BR91] MNN89] If the size ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. In Proc. 30. IEEE Symposium on Foundations of Computer Science, pages 54--59, 1989.

....NP hard [6] A recent result of Lund and Yannakakis [13] shows that this problem cannot be approximated in P with ratio c log 2 n for any c 1=4 unless NP = DTIME(n O(1) However there is known a polynomial time algorithm that finds a logarithmicfactor approximation. Recently Berger et al. [3] has been proved the following lemma. Lemma 5.9 For any 0, there is an NC algorithm for the weighted set cover problem that runs in O(log 4 n log m log 2 (nm) 6 ) time, uses O(n P m i=1 j Y i j) processors, and produces a cover of weight at most (1 ) log n times the weight of a ....

B. Berger, J. Rompel, and P. W. Shor, Efficient NC algorithms for set cover with applications to learning and geometry, in Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pp. 54--59, 1989. The full version is to appear in J. Algorithms.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., volume 30, pages 54--59, 1989.

....in NC as well. The reader is warned that this section is not self contained and is referred to the paper of [CF] as our description relies heavily on it. Anderson [An] was the first to notice the connection between set balancing and the hypergraph covering problems. Consequent to our results, [BRS] provided more efficient NC algorithms for these problems. We follow the notation of [CF] A hypergraph H = V; E) contains a finite set V of vertices of cardinality n and an edge set E of nonempty subsets of V . A subset T ae V of cardinality r is called an r sample. A frame F is a pair (H ; ....

B. Berger, J. Rompel and P. Shor, "Efficient NC algorithms for set cover with applications to learning and geometry", 30th Annual Symposium on the Foundations of Computer Science, 1989, pp. 2-7.

....the discretized version of R linear geometric concepts with random misclassification noise. A number of results [29, 30, 31, 26, 3, 28, 15, 13, 24, 14, 22, 12] have been obtained for geometric classes in Angluin s query learning model [1] as well. There has also been work on learning in parallel [36, 7, 37, 11, 4]. Of particular relevance is the work of Vitter and Lin [36, 37] They say that a concept class C is NC learnable (respectively, NC MC j learnable) if there exists a PAC learning algorithm for C in RNC that runs in polylogarithmic time with a polynomial number of processors on an arithmetic ....

....linearly separable functions (for constant d) simple k gons (k constant) unions of s axis parallel rectangles in the plane 6 . Furthermore they prove that the class of axisparallel rectangles, linear separators and simple k gons are NC MC j learnablefor j 1=2. Berger, Rompel and Shor [7] gave NC approximation algorithms for the unweighted and weighted set cover problems. They use these approximation algorithms to prove the NC learnability of concept classes formed by taking either finite unions or finite intersections of a fixed base class of finite VC dimension for which there ....

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Journal of Computer and System Sciences, 49(2):454--477, 1994.

....depend exponentially on d, the size of the target concept itself could be exponential in d. Thus, with the exception of some subclasses of C d s with polynomially sized concepts, our algorithm runs in time polynomial in the size of the target. Note, we can use parallel set covering techniques [10] to get our algorithm to run efficiently in parallel. A second contribution of our work is the conversion of our basic algorithm to a statistical query algorithm giving noise tolerance. Due to our covering approach, the statistical queries we make come from a modified distribution (that we ....

Bonnie Berger, John Rompel, and Peter W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Journal of Computer and System Sciences, 49(2):454--477, 1994.

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. B. Berger, J. Rompel, P. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Proc. 1989 IEEE FOCS, 54-59.

No context found.

B. Berger, J. Rompel, P. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Proc. 30th Symp. on Foundations of Computer Science, IEEE, 54-59(1989).

No context found.

B. Berger, J. Rompel, P. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Proc. 30th Symp. on Foundations of Computer Science, IEEE, 54-59(1989).

No context found.

B. Berger, J. Rompel and P. Shor, Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry, Journal of Computer and System Sciences 49, 1994, 454--477.

No context found.

B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Journal of Computer and System Sciences, 49(2):454--477, 1994.

No context found.

) Berger, B., Rompel, J., and Shor, P., "Efficient NC Algorithms for Set Cover with Application to Learning and Geometry," in Proc. of the 30th Annual Symposium on Foundations of Computer Science, pp. 54-59, 1989.

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B.Berger, J.Rompel and P.Shor, Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry, Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, (1989) 54-59.

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