Results 1  10
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221
Some optimal inapproximability results
, 2002
"... We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for ..."
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Cited by 751 (11 self)
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We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for MaxE2Sat, MaxCut, MaxdiCut, and Vertex cover. Warning: Essentially this paper has been published in JACM and is subject to copyright restrictions. In particular it is for personal use only.
The Budgeted Maximum Coverage Problem
, 1997
"... The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S 0 ` S such that the total cost of sets in S 0 does not exceed L, and the total weight of elements covered by S 0 is maxim ..."
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Cited by 188 (7 self)
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The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S 0 ` S such that the total cost of sets in S 0 does not exceed L, and the total weight of elements covered by S 0 is maximized. This problem is NPhard. For the special case of this problem, where each set has unit cost, a (1 \Gamma 1 e )approximation is known. Yet, no approximation results are known for the general cost version. The contribution of this paper is a (1 \Gamma 1 e )approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP ` DT IME(n log log n ). 1 Introduction The budgeted maximum coverage problem is defined as follows. A collection of sets S = fS 1 ; S 2 ; : : : ; Sm g with associated costs fc i g m i=1 is defined over a domain of elements X = fx 1 ; x 2 ; : : : ; x n g with associated weights fw i ...
The primaldual method for approximation algorithms and its application to network design problems.
, 1997
"... Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the prim ..."
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Cited by 137 (5 self)
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Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the primaldual method for approximation algorithms. We show how this technique can be used to derive approximation algorithms for a number of different problems, including network design problems, feedback vertex set problems, and facility location problems.
Algorithmic construction of sets for krestrictions
 ACM TRANSACTIONS ON ALGORITHMS
, 2006
"... This work addresses krestriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of kwise demands. For every k positions and every demand, there must be at least one string in the list that satis ..."
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Cited by 73 (2 self)
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This work addresses krestriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of kwise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science. The standard approach for deterministically solving such problems is via almost kwise independence or kwise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor, Schulman and Srinivasan [18]. Among other results, we greatly enhance the combinatorial objects in the heart of their method, called splitters, and construct multiway splitters, using a new discrete version of the topological Necklace Splitting Theorem [1]. We utilize our methods to show improved constructions for group testing [19] and generalized hashing [3], and an improved inapproximability result for SetCover under the assumption P != NP.
PolynomialTime Data Reduction for DOMINATING SET
 Journal of the ACM
, 2004
"... Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achiev ..."
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Cited by 64 (8 self)
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Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.
An online algorithm for maximizing submodular functions
, 2007
"... We present an algorithm for solving a broad class of online resource allocation jobs arrive one at a time, and one can complete the jobs by investing time in a number of abstract activities, according to some schedule. We assume that the fraction of jobs completed by a schedule is a monotone, submod ..."
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Cited by 59 (12 self)
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We present an algorithm for solving a broad class of online resource allocation jobs arrive one at a time, and one can complete the jobs by investing time in a number of abstract activities, according to some schedule. We assume that the fraction of jobs completed by a schedule is a monotone, submodular function of a set of pairs (v, τ), where τ is the time invested in activity v. Under this assumption, our online algorithm performs nearoptimally according to two natural metrics: (i) the fraction of jobs completed within time T, for some fixed deadline T> 0, and (ii) the average time required to complete each job. We evaluate our algorithm experimentally by using it to learn, online, a schedule for allocating CPU time among solvers entered in the 2007 SAT solver competition. 1
On the Hardness of Approximating Spanners
 Algorithmica
, 1999
"... A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns ..."
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Cited by 58 (14 self)
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A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k, approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k = 2, and the case k 5. Department of Computer Science, The Open University, 16 Klauzner st., Ramat Aviv, Israel, guyk@shaked.openu.ac.il. 1 Introduction The concept of graph spanners has been studied in several recent papers in the context of communication networks, distributed computing, robotics and computational geometry [ADDJ90, C94, CK94,...
Approximation Algorithms for Maximization Problems arising in Graph Partitioning
, 1998
"... Given a graph G = (V; E), a weight function w : E ! R + and a parameter k we examine a family of maximization problems arising naturally when considering a subset U ` V of size exactly k. Specifically we consider the problem of finding a subset U ` V of size k that maximizes : MaxkVertex Cover ..."
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Cited by 55 (5 self)
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Given a graph G = (V; E), a weight function w : E ! R + and a parameter k we examine a family of maximization problems arising naturally when considering a subset U ` V of size exactly k. Specifically we consider the problem of finding a subset U ` V of size k that maximizes : MaxkVertex Cover : the weight of edges incident with vertices in U . MaxkDense Subgraph : the weight of edges in the subgraph induced by U . MaxkCut : the weight of edges cut by the partition (U; V n U ). MaxkNot Cut : the weight of edges not cut by the partition (U; V n U ). We present a number of approximation algorithms based on linear and semidefinite programming, and obtain approximation ratios higher than those previously published.
Approximate Clustering without the Approximation
"... Approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much research effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as kmedian, kmeans, and minsum clustering. This quest for ..."
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Cited by 53 (17 self)
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Approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much research effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as kmedian, kmeans, and minsum clustering. This quest for better approximation algorithms is further fueled by the implicit hope that these better approximations also give us more accurate clusterings. E.g., for many problems such as clustering proteins by function, or clustering images by subject, there is some unknown “correct” target clustering and the implicit hope is that approximately optimizing these objective functions will in fact produce a clustering that is close (in symmetric difference) to the truth. In this paper, we show that if we make this implicit assumption explicit—that is, if we assume that any capproximation to the given clustering objective F is ǫclose to the target—then we can produce clusterings that are O(ǫ)close to the target, even for values c for which obtaining a capproximation is NPhard. In particular, for kmedian and kmeans objectives, we show that we can achieve this guarantee for any constant c> 1, and for minsum objective we can do this for any constant c> 2. Our results also highlight a somewhat surprising conceptual difference between assuming that the optimal solution to, say, the kmedian objective is ǫclose to the target, and assuming that any approximately optimal solution is ǫclose to the target, even for approximation factor say c = 1.01. In the former case, the problem of finding a solution that is O(ǫ)close to the target remains computationally hard, and yet for the latter we have an efficient algorithm.