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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 770 (38 self)
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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 probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits 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 vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an Nvertex graph to within a factor of N ɛ is NPhard.
APPROXIMATION ALGORITHMS FOR SCHEDULING UNRELATED PARALLEL MACHINES
, 1990
"... We consider the following scheduling problem. There are m parallel machines and n independent.jobs. Each job is to be assigned to one of the machines. The processing of.job j on machine i requires time Pip The objective is to lind a schedule that minimizes the makespan. Our main result is a polynomi ..."
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Cited by 241 (6 self)
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We consider the following scheduling problem. There are m parallel machines and n independent.jobs. Each job is to be assigned to one of the machines. The processing of.job j on machine i requires time Pip The objective is to lind a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worstcase ratio less than ~ unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.
The tracker: a threat to statistical database security
 ACM Trans. Database Syst
, 1979
"... The query programs of certain databases report raw statistics for query sets, which are groups of records specified implicitly by a characteristic formula. The raw statistics include query set size and sums of powers of values in the query set. Many users and designers believe that the individual re ..."
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Cited by 61 (4 self)
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The query programs of certain databases report raw statistics for query sets, which are groups of records specified implicitly by a characteristic formula. The raw statistics include query set size and sums of powers of values in the query set. Many users and designers believe that the individual records will remain confidential as long as query programs refuse to report the statistics of query sets which are too small. It is shown that the compromise of small query sets can in fact almost always be accomplished with the help of characteristic formulas called trackers. Schlorer’s individual tracker is reviewed, it is derived from known characteristics of a given individual and permits deducing additional characteristics he may have. The general tracker is introduced: It permits calculating statistics for arbitrary query sets, without requiring preknowledge of anything in the database. General trackers always exist if there are enough distinguishable classes of individuals in the database, in which case the trackers have a simple form. Almost all databases have a general tracker, and general trackers are almost always easy to find. Security is not guaranteed by the lack of a general tracker.
A Tabu Search Approach to Task Scheduling on Heterogeneous Processors under Precedence Constraints
, 1994
"... Parallel programs may be represented as a set of interrelated sequential tasks. When multiprocessors are used to execute such programs, the parallel portion of the application can be speeded up by an appropriate allocation of processors to the tasks of the application. Given a parallel application d ..."
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Cited by 42 (9 self)
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Parallel programs may be represented as a set of interrelated sequential tasks. When multiprocessors are used to execute such programs, the parallel portion of the application can be speeded up by an appropriate allocation of processors to the tasks of the application. Given a parallel application defined by a task precedence graph, the goal of task scheduling (or processor assignment) is thus the minimization of the makespan of the application. In a heterogeneous multiprocessor system, task scheduling consists in determining which tasks will be assigned to each processor, as well as the execution order of the tasks assigned to each processor. In this work, we apply the tabu search metaheuristic to the solution of the task scheduling problem on a heterogeneous multiprocessor environment under precedence constraints. The topology of the Mean Value Analysis solution package for product form queueing networks is used as the framework for performance evaluation. We show that tabu search ob...
Polynomial time approximation algorithms for machine scheduling: Ten open problems
 Journal of Scheduling
, 1999
"... We discuss what we consider to be the ten most vexing open questions in the area of polynomial time approximation algorithms for NPhard deterministic machine scheduling
problems. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature.
..."
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Cited by 40 (2 self)
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We discuss what we consider to be the ten most vexing open questions in the area of polynomial time approximation algorithms for NPhard deterministic machine scheduling
problems. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature.
Polynomial Time Approximation Schemes for ClassConstrained Packing Problems
 Proc. of Workshop on Approximation Algorithms
, 1999
"... . We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colors) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items ..."
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Cited by 32 (6 self)
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. We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colors) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can hold. In the classconstrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the classconstrained binpacking (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 01 classconstrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colors of items depends on the input size. Finally, we introduce the generalized classconstrained packing problem (GCCP), where each item may have more than one color. We show that GCCP is APX...
To weight or not to weight: where is the question?
 Proceedings of the 4th IEEE Israel Symposium on Theory of Computing and Systems
, 1996
"... We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) definition of niceness, we show that if a nice weighted problem is hard to approximate within r, then its polynomially bounded ..."
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Cited by 27 (7 self)
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We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) definition of niceness, we show that if a nice weighted problem is hard to approximate within r, then its polynomially bounded weighted version is hard to approximate within r \Gamma o(1). Then we turn our attention to specific problems, and we show that the unweighted
NearOptimal Dynamic Task Scheduling of Independent CoarseGrained Tasks onto a Computational Grid
 Proc. ICPP 2003
, 2003
"... problems is makespan. However, on a computational grid, the 2nd optimal makespan may be much longer than the optimal makespan because the computing power of a grid varies over time. So, if the performance measure is makespan, there is no approximation algorithm in general for scheduling onto a grid. ..."
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Cited by 25 (1 self)
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problems is makespan. However, on a computational grid, the 2nd optimal makespan may be much longer than the optimal makespan because the computing power of a grid varies over time. So, if the performance measure is makespan, there is no approximation algorithm in general for scheduling onto a grid. In this paper, a novel criterion of a schedule is proposed. The proposed criterion is called total processor cycle consumption, which is the total number of instructions the grid could compute until the completion time of the schedule. Moreover, for the criterion, this paper gives a (1 + )approximation algorithm for scheduling n independent coarsegrained tasks with the same length onto a grid with m processors. The proposed algorithm does not use any prediction information on the performance of underlying resources. This result implies a nontrivial result that the computing power consumed by a parameter sweep application can be limited in such a case within (1 + ) times that required by an optimal schedule, regardless how the speed of each processor varies over time.
A Note on Complexity of Lp Minimization
, 2010
"... We show that the Lp (0 ≤ p < 1) minimization problem arising from sparse solution construction and compressed sensing is both hard and easy. More precisely, for any fixed 0 < p < 1, we prove that checking the global minimal value of the problem is NPHard; but computing a local minimizer of ..."
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Cited by 25 (3 self)
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We show that the Lp (0 ≤ p < 1) minimization problem arising from sparse solution construction and compressed sensing is both hard and easy. More precisely, for any fixed 0 < p < 1, we prove that checking the global minimal value of the problem is NPHard; but computing a local minimizer of the problem is polynomialtime doable. We also develop an interiorpoint algorithm with a provable complexity bound and demonstrate preliminary computational results of effectiveness of the algorithm.