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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 797 (39 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.
Hedging uncertainty: Approximation algorithms for stochastic optimization problems
 In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization
, 2004
"... We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, ..."
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Cited by 77 (13 self)
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We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primaldual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1
A polynomial time approximation scheme for the multiple knapsack problem
 SIAM J. COMPUT
, 2006
"... The multiple knapsack problem (MKP) is a natural and wellknown generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to f ..."
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Cited by 40 (0 self)
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The multiple knapsack problem (MKP) is a natural and wellknown generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the generalized assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APXhard and a 2approximation, for it is implicit in the work of Shmoys and Tardos [Math. Program. A, 62 (1993), pp. 461–474], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme (PTAS) for MKP. Apart from its inherent theoretical interest as a common generalization of the wellstudied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APXhard. We substantiate this by showing that slight generalizations of MKP are APXhard. Thus our results help demarcate the boundary at which instances of GAP become APXhard. An interesting aspect of our approach is a PTASpreserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.
Resource Allocation Algorithms for Virtualized Service Hosting Platforms
, 2010
"... Commodity clusters are used routinely for deploying service hosting platforms. Due to hardware and operation costs, clusters need to be shared among multiple services. Crucial for enabling such shared hosting platforms is virtual machine (VM) technology, which allows consolidation of hardware resour ..."
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Cited by 29 (4 self)
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Commodity clusters are used routinely for deploying service hosting platforms. Due to hardware and operation costs, clusters need to be shared among multiple services. Crucial for enabling such shared hosting platforms is virtual machine (VM) technology, which allows consolidation of hardware resources. A key challenge, however, is to make appropriate decisions when allocating hardware resources to service instances. In this work we propose a formulation of the resource allocation problem in shared hosting platforms for static workloads with servers that provide multiple types of resources. Our formulation supports a mix of besteffort and QoS scenarios, and, via a precisely defined objective function, promotes performance, fairness, and cluster utilization. Further, this formulation makes it possible to compute a bound on the optimal resource allocation. We propose several classes of resource allocation algorithms, which we evaluate in simulation. We are able to identify an algorithm that achieves average performance close to the optimal across many experimental scenarios. Furthermore, this algorithm runs in only a few seconds for large platforms and thus is usable in practice.
The approximability of NPhard problems
 In Proceedings of the Annual ACM Symposium on Theory of Computing
, 1998
"... Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical” ..."
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Cited by 17 (0 self)
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Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical”
FULLY DYNAMIC ALGORITHMS FOR BIN PACKING: BEING (MOSTLY) MYOPIC HELPS
, 1998
"... The problem of maintaining an approximate solution for onedimensional bin packing when items may arrive and depart dynamically is studied. In accordance with various work on fully dynamic algorithms, and in contrast to prior work on bin packing, it is assumed that the packing may be arbitrarily re ..."
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Cited by 13 (0 self)
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The problem of maintaining an approximate solution for onedimensional bin packing when items may arrive and depart dynamically is studied. In accordance with various work on fully dynamic algorithms, and in contrast to prior work on bin packing, it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items. In this context our main result is a fully dynamic approximation algorithm for bin packing MMP that is 5/4competitive and requires Θ(log n) time per operation (i.e., for an Insert or a Delete of an item). This competitive ratio of 5 is nearly as good as that of the best practical offline algorithms. Our algorithm utilizes 4 the technique (introduced here) whereby the packing of an item is done with a total disregard for already packed items of a smaller size. This myopic packing of an item may then cause several smaller items to be repacked (in a similar fashion). With a bit of additional sophistication to avoid certain “bad” cases, the number of items (either individual items or “bundles” of very small items treated as a whole) that needs to be repacked is bounded by a constant.
Vector assignment problems: A general framework
 Journal of Algorithms
, 2002
"... We present a general framework for vector assignment problems. In such problems one aims at assigning n input vectors to m machines such that the value of a given target function is minimized. While previous approaches concentrated on simple target functions such as maxmax, the general approach pre ..."
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Cited by 7 (2 self)
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We present a general framework for vector assignment problems. In such problems one aims at assigning n input vectors to m machines such that the value of a given target function is minimized. While previous approaches concentrated on simple target functions such as maxmax, the general approach presented here enables us to design a polynomial time approximation scheme (PTAS) for a wide class of target functions. In particular, thanks to a novel technique of preprocessing the input vectors, we are able to deal with nonmonotone target functions. Such target functions arise in vector assignment problems in the context of video transmission and broadcasting.
A survey on approximation algorithms for scheduling with machine unavailability
 In Algorithmics of Large and Complex Networks: Design, Analysis, and Simulation
, 2009
"... Abstract. In this chapter we present recent contributions in the field of sequential job scheduling on network machines which work in parallel; these are subject to temporary unavailability. This unavailability can be either unforeseeable (online models) or known a priori (offline models). For the o ..."
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Cited by 5 (0 self)
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Abstract. In this chapter we present recent contributions in the field of sequential job scheduling on network machines which work in parallel; these are subject to temporary unavailability. This unavailability can be either unforeseeable (online models) or known a priori (offline models). For the online models we are mainly interested in preemptive schedules for problem formulations where the machine unavailability is given by a probabilistic model; objectives of interest here are the sum of completion times and the makespan. Here, the nonpreemptive case is essentially intractable. For the offline models we are interested in nonpreemptive schedules where we consider the makespan objective; we present approximation algorithms which are complemented by suitable inapproximability results. Here, the preemptive model is polynomialtime solvable for large classes of settings. 1
Faster Approximation Algorithms for Scheduling with Fixed Jobs
"... We study the problem of scheduling jobs on identical parallel machines without preemption. In the considered setting, some of the jobs are already assigned machines and starting times, for example due to external constraints not explicitly modelled. The objective is to assign the rest of the jobs in ..."
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Cited by 4 (1 self)
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We study the problem of scheduling jobs on identical parallel machines without preemption. In the considered setting, some of the jobs are already assigned machines and starting times, for example due to external constraints not explicitly modelled. The objective is to assign the rest of the jobs in order to minimize the makespan. It is known that this problem cannot be approximated better than within a factor of 3/2 unless P = NP. An algorithm that achieves 3/2 + ɛ for any ɛ> 0 was presented by Diedrich and Jansen [DJ09], but its running time is doubly exponential in 1/ɛ. We present an improved algorithm with approximation ratio 3/2 and polynomial running time. We also give matching results for the related problem of scheduling with reservations. The new algorithm is both faster and conceptually simpler than the previously known algorithms. 1