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Asymmetric kcenter is log ∗ nhard to Approximate
 In Proc. SODA
, 2005
"... In the Asymmetric kCenter problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality. The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum ..."
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Cited by 34 (4 self)
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distance of any point to its center is as small as possible. We show that the Asymmetric kCenter problem is hard to approximate up to a factor of log ∗ n − Θ(1) unless NP ⊆ DTIME(n log log n). Since an O(log ∗ n)approximation algorithm is known for this problem, this essentially resolves
Lecture 7: Asymmetric KCenter
, 2007
"... In this lecture, we will consider the Kcenter problem, both in its symmetric and asymmetric variants. For this lecture, recall that a metric space (V, d) is consists of a set of Points V along with a function d: V × V → R + which satisfies the following properties: • ∀x ∈ V, d(x, x) = 0 • (Triangl ..."
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In this lecture, we will consider the Kcenter problem, both in its symmetric and asymmetric variants. For this lecture, recall that a metric space (V, d) is consists of a set of Points V along with a function d: V × V → R + which satisfies the following properties: • ∀x ∈ V, d(x, x) = 0
Asymmetry in kCenter Variants
, 2003
"... This paper explores three concepts: the kcenter problem, some of its variants, and asymmetry. The kcenter problem is a fundamental clustering problem, similar to the kmedian problem. Variants of kcenter may more accurately model reallife problems than the original formulation. Asymmetry is a ..."
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Cited by 3 (2 self)
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significant impediment to approximation in many graph problems, such as kcenter, facility location, kmedian and the TSP. We demonstrate
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 778 (5 self)
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o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 822 (39 self)
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in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNPhard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include
Approximate Signal Processing
, 1997
"... It is increasingly important to structure signal processing algorithms and systems to allow for trading off between the accuracy of results and the utilization of resources in their implementation. In any particular context, there are typically a variety of heuristic approaches to managing these tra ..."
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Cited by 516 (2 self)
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these tradeoffs. One of the objectives of this paper is to suggest that there is the potential for developing a more formal approach, including utilizing current research in Computer Science on Approximate Processing and one of its central concepts, Incremental Refinement. Toward this end, we first summarize a
Two O(log k) approximation algorithms for the asymmetric kcenter problem
 Proceedings of the 8th Conference on Integer Programming and Combinatorial Optimization
, 2001
"... Given a set V of n points and the distances between each pair, the kcenter problem asks us to choose a subset C \subset V of size k that minimizes the maximum over all points of the distance from C to the point. This problem is NP hard even when the distances are symmetric and satisfy the triangle ..."
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Given a set V of n points and the distances between each pair, the kcenter problem asks us to choose a subset C \subset V of size k that minimizes the maximum over all points of the distance from C to the point. This problem is NP hard even when the distances are symmetric and satisfy the triangle
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 1231 (13 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds
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