P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538{ 547, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....P 1 and P 2 are 3t and t bits long. For more details on use of two prover protocols in inapproximability results, see the paper by Hastad [21] Lemma 4. 5 (Bellare and Rogoway [8] Using the two prover protocol described above, one can describe a reduction T 1 from 3SAT formulas of size m to QP [2 ] such that: 1. If # is satisfiable, then OPT (T 1 (#) w 1 for some w 1 . 2. If # is unsatisfiable, then OPT (T 2 (#) w 2 for some w 2 . 3. f for some fixed constant f 1. Moreover, this reduction runs in time 2 . From Quadratic Programming to Norm Maximization We use the ....

....construction of Brieden. Brieden describes a sequence of interesting reductions that converts an instance of quadratic programming to an instance of the norm maximization problem. Lemma 4. 6 (Brieden [13] There is a reduction T 2 from Quadratic Programming to the Norm Maximization that maps QP [2 ] into NM[2 ] with the following property: for any input L of QP to T 2 and for any # 0, OPT (L) 1 #) # OPT (T 2 (L) 1 #)OPT (L) Moreover, the reduction T 2 runs in time 2 and the inequalities in T 2 (L) describe a centrally symmetric polytope, that is, a polytope P ....

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P.K. Agarwal and C.M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538--547, 2000.

....in the literature. For example, Charikar et al. CC 97] study an incremental version of the clustering problem for minimizing the maximum radius. Pferschy et al. PRW94] study geometric versions of clustering problems using objectives such as minimizing the total perimeter. Agarwal and Procopiuc [AP00] study projective clustering problems where the goal is to cover a set of points in d using hyper strips, and the objective is to minimize the maximum width of the strips. References where other types of clustering problems are studied include [Ma99,ABC 98,GH98,DKS97,Da94,BKK94] 2 ....

P. K. Agarwal and C. M. Procopiuc, "Approximation Algorithms for Projective Clustering", Proc. 11th ACM-SIAM Symposium on Discrete Algorithms (SODA'2000), San Francisco, CA, Jan. 2000, pp. 538--547.

....to decide whether a set of n points in the plane can be covered by k lines. This immediately implies that projective clustering is NP Complete even in the planar case. In fact, it also implies that approximating the minimum width within a constant factor is NP Complete. Agarwal and Procopiuc [2] propose an algorithm with near linear running time that computes a cover by O(k log k) strips of width no larger than the width of the optimal cover by k strips. The algorithm extends to covering points by hyper cylinders in R d and to a few special cases of covering points by hyper strips in ....

....the strip subsets of S to be the (not necessarily disjoint) sets S i = S oe i . For a strip oe, we call a pair of points p; q 2 S oe an anchor pair of oe if d(p; q) 1 The base of all logarithms is 2, unless otherwise specified. 2 diam(S oe) 2: The following lemma was proved in [2]. We repeat the proof here as it will be useful later on. Lemma 2.1. Let oe 2 Sigma , and let (p; q) be an anchor pair of oe . Then there exists a point r 2 S so that oe(p; q; r) covers all points of S oe and d(r; pq ) 3w . Proof: Let w w be the width of oe , S = ....

P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538--547, 2000.

....X repeat choose a random H 2 X r ; if H r 6= for all r 2 R then return H choose a r 2 R s.t. r H = if (r) X) 2k then successful : successful 1 for all x 2 r do (x) 2 (x) until successful : 12k log n return no Figure 7: A randomized hitting set al..gorithm. cases [6, 8], and we will mention them below. 6 Facility Location Problems A typical facility location problem is de ned as follows: Given a set D = fd 1 ; d n g of n demand points in R d , a parameter p, and a distance function , we wish to nd a set S of p supply objects (points, lines, ....

....whether w = 0 (i.e, D can be covered by p lines) is NP Hard [145] which not only proves that the p line center problem is NP Complete, but also proves that approximating w within a constant factor is NP Complete. Since even approximating w is NP Hard, Agarwal and Procopiuc [8] developed an ecient algorithm for the case in which one approximates both w and p. In particular, let w p = w p (D) denote the size of the Euclidean p line center of S. By modifying the hitting set al..gorithm described in Section 5, they presented a randomized algorithm that computes ....

P. K. Agarwal and C. M. Procopiuc, Approximation algorithms for projective clustering, Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, 2000, pp. 538-547.

....implicitly, then T (n) is proportional to P r2R jrj. But R could be quite large and in many geometric settings it is defined explicitly. In such cases the main difficulty is to run the above algorithm efficiently without computing R explicitly. Such algorithms have been proposed in some cases [6, 8], and we will mention them below. 6 Facility Location Problems A typical facility location problem is defined as follows: Given a set D = fd 1 ; d n g of n demand points in R d , a parameter p, and a distance function ffi, we wish to find a set S of p supply objects (points, lines, ....

....is NP Complete, which not only proves that the p line center Geometric Optimization June 6, 2000 Facility Location Problems 18 problem is NP Complete, but also proves that approximating w within a constant factor is NP Complete. Since even approximating w is NP Complete, Agarwal and Procopiuc [8] developed an efficient algorithm for the case in which one approximates both w and p. In particular, let w p = w p (D) denote the size of the Euclidean p line center of S. By modifying the hitting set al..gorithm described in Section 5, they presented a randomized algorithm that computes O(p log ....

P. K. Agarwal and C. M. Procopiuc, Approximation algorithms for projective clustering, Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, 2000, pp. 538--547.

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P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538{ 547, 2000.

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P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538{ 547, 2000.

No context found.

P. K. Agarwal and C. Procopiuc. Approximation algorithms for projective clustering. Proc. SODA, pages 538--547, 2000.

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P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538{ 547, 2000.

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P. K. Agarwal and C. M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538{ 547, 2000.

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