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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.
A PTAS for the Multiple Knapsack Problem
, 1993
"... The Multiple Knapsackproblem (MKP) is a natural and well known 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 fin ..."
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Cited by 113 (2 self)
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The Multiple Knapsackproblem (MKP) is a natural and well known 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 [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme 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 that are very restricted versions of GAP are APXhard. Thus our results help demarcate the boundary at which instances of GAP becomeAPXhard. An interesting and novel aspect of our approach is an approximation preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.
Nearoptimal hardness results and approximation algorithms for edgedisjoint paths and related problems
 Journal of Computer and System Sciences
, 1999
"... We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjo ..."
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Cited by 108 (12 self)
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We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NPhard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)approximation algorithm for it. For directed networks, we show that even the single sourcesink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.
On Multidimensional Packing Problems
, 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 98 (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 multiple resource requirements. We obtain a
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and acce ..."
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Cited by 65 (8 self)
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The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently
On Approximation Properties of the Independent Set Problem for Degree 3 Graphs
 In Proc. of Workshop on Algorithms and Data Structures
, 1995
"... . The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNPcomplete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNPcomplete at the lowest possible degree bounds. N ..."
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Cited by 48 (0 self)
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. The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNPcomplete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNPcomplete at the lowest possible degree bounds. Next we study better polytime approximation of the problem for degree 3 graphs, and improve the previously best ratio, 5 4 , to arbitrarily close to 6 5 . This result also provides improved polytime approximation ratios, B+3 5 + ffl, for odd degree B. 1 Introduction The area of efficient approximation algorithms for NPhard optimization problems has recently seen dramatic progress with a sequence of breakthrough achievements. Even when restricted only to the area of constant bound approximation the following remarkable results have been obtained in the last few years. The subclass of NP optimization problems, called MAX SNP, consisting solely of constant ratio approximable problems ...
Nonapproximability results for scheduling problems with minsum criteria
 Proceedings of the 6th International IPCO Conference on Integer Programming and Combinatorial Optimization
, 1998
"... We provide several nonapproximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless P = NP, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are ..."
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Cited by 44 (3 self)
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We provide several nonapproximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless P = NP, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by APXhardness proofs.
We show that, whereas scheduling on unrelated machines with unit weights is polynomially solvable, the problem becomes APXhard if release dates or weights are added. We further show APXhardness for scheduling in flow shops, job shops, and open shops. We also investigate the problems of scheduling on parallel machines with precedence constraints and unit processing times, and two variants of the latter problem with unit communication delays; for these problems we provide lower bounds on the worstcase behavior of any
polynomialtime approximation algorithm through the gap reduction technique.
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.
ON THE COMPLEXITY OF APPROXIMATING kSET PACKING
 COMPUTATIONAL COMPLEXITY
, 2006
"... Given a kuniform hypergraph, the Maximum kSet Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of Ω(k / ln k) unless P = NP. This improves the previous hardness of approximation factor of k/2 O( √ ln k) ..."
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Cited by 35 (0 self)
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Given a kuniform hypergraph, the Maximum kSet Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of Ω(k / ln k) unless P = NP. This improves the previous hardness of approximation factor of k/2 O( √ ln k) by Trevisan. This result extends to the problem of kDimensionalMatching.
Bin packing in multiple dimensions: Inapproximability results and approximation schemes
 MATHEMATICS OF OPERATIONS RESEARCH
, 2006
"... We study the multidimensional generalization of the classical Bin Packing problem: Given a collection of ddimensional rectangles of specified sizes, the goal is to pack them into the minimum number of unit cubes. A long history of results exists for this problem and its special cases. Currently, t ..."
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Cited by 30 (7 self)
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We study the multidimensional generalization of the classical Bin Packing problem: Given a collection of ddimensional rectangles of specified sizes, the goal is to pack them into the minimum number of unit cubes. A long history of results exists for this problem and its special cases. Currently, the best known approximation algorithm for packing twodimensional rectangles achieves a guarantee of 1.69 in the asymptotic case (i.e., when the optimum uses a large number of bins) [3]. An important open question has been whether 2−dimensional bin packing is essentially similar to the 1−dimensional case in that it admits an asymptotic polynomial time approximation scheme (APTAS) [12, 17] or not. We answer the question in the negative and show that the problem is APX hard in the asymptotic sense. On the positive side, we give the following results: First, we consider the special case where we have to pack ddimensional cubes into the minimum number of unit cubes. We give an asymptotic polynomial time approximation scheme for this problem. This represents a significant improvement over the previous best known asymptotic approximation factor of 2 − (2/3) d [21] (1.45 for d = 2 [11]), and settles the approximability of the problem. Second, we give a polynomial time algorithm for packing arbitrary rectangles into at most OPT square bins with sides of length 1 + ε, where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result does not have an additive constant term i.e., is not an asymptotic result. As a corollary, we obtain a polynomial time approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle, settling also the approximability of this problem.