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29
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 626 (6 self)
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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 NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma 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 .
A Tight Analysis of the Greedy Algorithm for Set Cover
, 1995
"... We establish significantly improved bounds on the performance of the greedy algorithm for approximating set cover. In particular, we provide the first substantial improvement of the 20 year old classical harmonic upper bound, H(m), of Johnson, Lovasz, and Chv'atal, by showing that the performance ra ..."
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Cited by 88 (0 self)
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We establish significantly improved bounds on the performance of the greedy algorithm for approximating set cover. In particular, we provide the first substantial improvement of the 20 year old classical harmonic upper bound, H(m), of Johnson, Lovasz, and Chv'atal, by showing that the performance ratio of the greedy algorithm is, in fact, exactly ln m \Gamma ln ln m+ \Theta(1), where m is the size of the ground set. The difference between the upper and lower bounds turns out to be less than 1:1. This provides the first tight analysis of the greedy algorithm, as well as the first upper bound that lies below H(m) by a function going to infinity with m. We also show that the approximation guarantee for the greedy algorithm is better than the guarantee recently established by Srinivasan for the randomized rounding technique, thus improving the bounds on the integrality gap. Our improvements result from a new approach which might be generally useful for attacking other similar problems. ...
On Multidimensional Packing Problems
 TENTH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1999
"... We study the approximability of multidimensional generalizations of the classical problems of multiprocessor scheduling, bin packing and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integ ..."
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Cited by 66 (4 self)
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We study the approximability of multidimensional generalizations of the classical problems of multiprocessor scheduling, bin packing and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n ddimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension across all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. We obtain a variety of new algorithmic as well as inapproximability results for these problems. For vector scheduling, we give a PTAS when d is a fixed constant, and an O(minflog dm; log 2 dg)approximation in gen...
The price of being nearsighted
 In SODA ’06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
, 2006
"... Achieving a global goal based on local information is challenging, especially in complex and largescale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible tradeoff between the amount of local information and the quality o ..."
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Cited by 59 (13 self)
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Achieving a global goal based on local information is challenging, especially in complex and largescale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible tradeoff between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρ∆) 1/kapproximation for general covering and packing problems in time O(k 2), where ρ depends on the LP’s coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n 1/k) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its kneighborhood. 1
Approximation algorithms for disjoint paths and related routing and packing problems
 Mathematics of Operations Research
, 2000
"... Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems consi ..."
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Cited by 54 (1 self)
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Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multicommodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented. Key words and phrases. Disjoint paths, approximation algorithms, unsplittable flow, routing, packing, integer programming, multicommodity flow, randomized algorithms, rounding, linear programming. 1
An Extension of the Lovász Local Lemma, and its Applications to Integer Programming
 In Proceedings of the 7th Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mea ..."
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Cited by 31 (6 self)
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The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NPhard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1
The pipelined set cover problem
, 2003
"... Abstract. A classical problem in query optimization is to find the optimal ordering of a set of possibly correlated selections. We provide an abstraction of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the elements to be covered ..."
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Cited by 30 (5 self)
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Abstract. A classical problem in query optimization is to find the optimal ordering of a set of possibly correlated selections. We provide an abstraction of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the elements to be covered and the elements covered at each stage are discarded. We show that several natural heuristics for this NPhard problem, such as the greedy setcover heuristic and a localsearch heuristic, can be analyzed using a linearprogramming framework. These heuristics lead to efficient algorithms for pipelined set cover that can be applied to order possibly correlated selections in conventional database systems as well as datastream processing systems. We use our linearprogramming framework to show that the greedy and localsearch algorithms are 4approximations for pipelined set cover. We extend our analysis to minimize the lpnorm of the costs paid by the sets, where p ≥ 2 is an integer, to examine the improvement in performance when the total cost has increasing contribution from initial sets in the pipeline. Finally, we consider the online version of pipelined set cover and present a competitive algorithm with a logarithmic performance guarantee. Our analysis framework may be applicable to other problems in query optimization where it is important to account for correlations. 1
Path Coloring on the Mesh
 In Proc. of the 37th Annual IEEE Symposium on Foundations of Computer Science
, 1996
"... In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Rag ..."
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Cited by 29 (0 self)
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In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Raghavan and Upfal [22] as a model for routing in alloptical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N ) approximation result of Kleinberg and Tardos [14] for path coloring on the N \Theta N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N ) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sublogarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integral...
Adaptivity and Approximation for Stochastic Packing Problems
"... ... (PIP) — the problems of finding a maximumvalue 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), bmatching, and others ..."
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Cited by 28 (2 self)
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... (PIP) — the problems of finding a maximumvalue 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), bmatching, and others. PIP can also be seen as a “multidimensional” knapsack problem where we wish to pack a maximumvalue collection of items with vectorvalued sizes. In our stochastic setting, the vectorvalued size of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution. Following the framework of [3], we consider both adaptive and nonadaptive policies for solving such problems, adaptive policies having the flexibility of being able to make ...
Tree Layout for Internal Network Characterizations in Multicast Networks
 in 3rd International Workshop on Networked Group Communication (NGC
, 2001
"... There has been considerable activity recently to develop monitoring and debugging tools for a multicast session (tree). With these tools in mind, we focus on the problem of how to lay out multicast sessions so as to cover a set of links of interest within a network. We define three variations of t ..."
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Cited by 22 (0 self)
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There has been considerable activity recently to develop monitoring and debugging tools for a multicast session (tree). With these tools in mind, we focus on the problem of how to lay out multicast sessions so as to cover a set of links of interest within a network. We define three variations of this layout (cover) problem that differ in what it means for a link to be covered. We then focus on the identifiability problem, to determine whether a given set of candidate multicast trees can cover the set of links of interest; and the minimum cost problem, to determine the minimum cost set of trees that cover the links in question. We establish efficient algorithms to solve the identifiability problem and show that, with few exceptions, the minimum cost problems are NPhard and that even finding an approximation within a certain factor is NPhard. One exception is when the underlying network topology is a tree. For this case, we demonstrate an efficient algorithm that finds the optimal solution. We also present several computationally efficient heuristics and their evaluation through simulation. We find that two heuristics, a greedy heuristic that combines sets of trees with three or fewer receivers, and a heuristic based on generalizing our tree algorithm, both perform reasonably well. The remainder of the paper applies our techniques to the vBNS and Abilene networks, examining the effectiveness of the different heuristics and the sensitivity of the costs to the choice of routing algorithm. 1