M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....true) 26 Both the satisfiability problem (or SAT problem for short) and the set covering problem is known to be NP hard [14] For the set covering problem the following result also holds, showing that it is NP hard to approximate this problem within every constant factor c 1. Theorem 19 [7] For every c 1 there is a polynomial time reduction that, given an instance of SAT, produces an instance of the set covering problem and a number K 2 N with the properties: if is satisfiable then there exists an exact cover of size K, but if is not satisfiable then every cover has size at ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell, Efficient multi-prover interactive proofs with applications to approximation problems, in Proceedings of the 25th ACM Symposium on the Theory of Computing, pp. 113-131, 1993.

....true) 32 Both the satisfiability problem (or SAT problem for short) and the set covering problem is known to be NP hard [15] For the set covering problem the following result also holds, showing that it is NP hard to approximate this problem within every constant factor c 1. Theorem 19 [7] For every c 1 there is a polynomial time reduction that, given an instance of SAT, produces an instance of the set covering problem and a number K 2 N with the properties: if is satisfiable then there exists an exact cover of size K, but if is not satisfiable then every cover has size at ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell, Efficient multi-prover interactive proofs with applications to approximation problems, in: Proceedings of the 25th ACM Symposium on the Theory of Computing (1993) 113--131.

....the geometric facts used in our reduction to SV1might be indicative of the nature of relevant techniques. A related open problem is to improve our results by proving the problems NP hard rather than almost NP hard. More efficient 2 prover, 1 round interactive proofs for NP (as conjectured in [BG ] or a direct reduction from [AL ] might help. Acknowledgements Thanks to Madhu Sudan for suggesting that techniques from interactive proofs might be helpful in proving the hardness of lattice vector problems, and to Ravi Kannan for his prompt responses to questions on lattices. We also thank ....

M. Bellare, S. Goldwasser, C. Lund, A. Russell. Efficient Multi-Prover Interactive Proofs with Applications to Approximation Problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....the same as in [3] Definition 4 (Set Cover) An instance of set cover consists of a ground set U and a collection of subsets S 1 ; Sm of U . A cover is a subcollection of the S i s whose union is U . The cover is said to be exact if the sets in the cover are pairwise disjoint. In [4], Bellare, Goldwasser et al. show that for every constant c 1 there is a polynomial time reduction that, on input an instance OE of SAT, produces an instance of set cover and an integer d with the following properties: ffl If OE is satisfiable, there is an exact cover of size d, ffl If OE is ....

....CVP is NP hard to approximate within any constant factor. Theorem 1 For every constant c 1 the promise problem GapCVP c is NPhard. Proof: Let c be a constant greater than one. We reduce SAT to GapCVP c . Let OE be an instance of SAT. Apply the reduction from Bellare, Goldwasser et al. [4] to the formula OE, to obtained instance of set cover U; S 1 ; Sm and integer k. Let n be the size of U and let S 2 f0; 1g n Thetam be the matrix defined by S i;j = 1 iff i 2 S j . Define N and y as follows: N = ffS I y = ff where ff is an integer such that ff ....

M. Bellare, S. Goldwasser, C. Lund, A. Russel, "Efficient Multi-Prover Interactive Proofs with Applications to Approximation Problems", In Proc. 25th ACM Symp. on Theory of Computing, 1993, pp. 113-131.

....of Proofs. The error probability of a PCP system is the maximal fraction of Psi that can be satisfied in case the input is not in L. Introducing the PCP scheme of characterizing NP has created an avalanche of hardness results for approximation problems ( FGL 91, AS92, ALM 92, LY94, BGLR93, BGS98, Has97] to mention only a few) For most of these applications, the characterization of NP with constant error probability and variables of constant School of Mathematical Sciences, Tel Aviv University, ISRAEL Weizmann Inst. of Science , ISRAEL range [AS92, ALM 92] suffices. In ....

....of constant School of Mathematical Sciences, Tel Aviv University, ISRAEL Weizmann Inst. of Science , ISRAEL range [AS92, ALM 92] suffices. In order to prove NP hardness of other problems however, sub constant errorprobability had turned out to be essential. For example [LY94] and [BGLR93] were able to prove approximating SET COVER to within logarithmic factors almost NP hard, using the constant error probability PCP characterization of NP. As to proving NP hardness for that problem, BGLR93] had suggested the sliding scale conjecture. The BGLR conjecture states that, even if ....

[Article contains additional citation context not shown here]

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....; mg such that S i2I C i = S. In this case the sets C i ; i 2 I , are called a cover of S. A cover is called exact if the sets in a cover are mutually disjoint. For the set covering problem the following result holds, showing that it is hard to approximate within every factor c 1: Theorem 7. [4] For every c 1 there is a polynomial time reduction that, given an instance of SAT, produces an instance of the set covering problem and a number K 2 N with the properties: if is satisfiable then there exists an exact cover of size K, if is not satisfiable then every cover has size at ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell, Efficient multi-prover interactive proofs with applications to approximation problems, in Proceedings of the 25th ACM Symposium on the Theory of Computing, pp. 113-131, 1993.

....of the complexity of approximation. There has been great progress in nailing down the exact approximation ratio for which approximation becomes hard. For instance, in the case of Clique, this ratio has gone from 2 log 1 Gammaffi n to n 1=3 ffl in four years [FGL 91, AS92, ALM 92, BGLR93, FK94, BS94, BGS95] Chromatic Number has seen similar progress [LY93b, KLS93, BS94, BGS95, F 94] In the case of Set Cover, we now even know a threshold result. Feige[Fei95b] improving upon the work of [LY93b, BGLR93] recently showed that achieving a ratio (1 Gamma ffl)ln n for Set Cover is ....

....1 Gammaffi n to n 1=3 ffl in four years [FGL 91, AS92, ALM 92, BGLR93, FK94, BS94, BGS95] Chromatic Number has seen similar progress [LY93b, KLS93, BS94, BGS95, F 94] In the case of Set Cover, we now even know a threshold result. Feige[Fei95b] improving upon the work of [LY93b, BGLR93] recently showed that achieving a ratio (1 Gamma ffl)ln n for Set Cover is hard for any fixed ffl 0, whereas a well known polynomial time algorithm ( Joh74, Lov75] achieves a ratio 1 ln n. But Set Cover is an anomaly; threshold results for most interesting optimization problems remain out ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM STOC, pp 113-- 131, 1993.

....in [BLR90] This new verifier is inefficient in the number of random bits it uses (this number is polynomial in the proof size) but it queries only O(1) bits from the proof. The verifier obtained through the above composition is (log n; 1) restricted. Since [ALM 92] a sequence of papers [PS, BGLR93, BS94, BGS95, H96] have constructed more and more efficient (in terms of constant factors) log n; 1) restricted verifiers for SAT. The latest verifier [H97] needs only 3 query bits 8 . Some of our ideas from Section 4.4 regarding the concatenation property were useful in achieving some of ....

....such verifiers for a log 1 Gammaffl n. It also generalizes our low degree test in an important way (see also [ASu97] Problems other than Clique and MAX SNP problems have also been shown hard to approximate. These include Chromatic Number and Set Cover (Lund and Yannakakis [LY94] see also [BGLR93, KLS93] and problems on lattices, codes, and linear systems (Arora, Babai, Stern and Sweedyk [ABSS93] Independently of these works, other hardness results for a variety of approximation problems were obtained by Bellare [Bel93] Bellare and Rogaway [BR93] and Zuckerman ( Zuc93] We refer the ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....of probabilistically checkable proofs (PCPs) Using this connection it became possible to prove hardness of approximation results for ff(G) under the assumption that P 6= NP, and even sharper hardness of approximation results under the assumption that NP 6 ZPP. After a long line of work [13, 4, 3, 6, 14, 7, 5, 21, 22], it has been shown hard to approximate ff(G) to within N 1 Gammaffl for any constant ffl 0. Lund and Yannakakis [28] reduced approximating ff(G) to approximating (G) thereby showing it hard to approximate (G) to within N c for some constant c 0. Subsequent work [24, 7, 16] has somewhat ....

....has small independent sets. The problem of approximating ff(G 0 ) within a constant factor is reduced to that of approximating ff(G) within a factor of O(N 1= 1 f) The quality of the hardness result depends on f , the number of amortized free bits in the PCP. Following a long line of work [6, 14, 7, 5, 21], Hastad [22] shows how to obtain f arbitrarily close to 0, implying that ff(G) is hard to approximate within a factor O(N 1 Gammaffl ) for any ffl 0. To prove hardness of approximation results for (G) a direct reduction from ff(G) to (G) was used. Known reductions do not preserve the ratio ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....as the most natural extension of the old characterization of NP [Coo71] if one has in mind proving hardness results for approximation problems. This characterization has already been used to obtain quite a few hardness results for approximation problems [FGL 91, AS92, ALM 92, PY91, LY94, BGLR93, KLS93, BGS95, Has96a, Has96b, Has97] The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM 92] seemed at first as the best possible up to constant factors. A stronger characterization was later conjectured in [BGLR93] one that, as an immediate outcome, implies ....

....[FGL 91, AS92, ALM 92, PY91, LY94, BGLR93, KLS93, BGS95, Has96a, Has96b, Has97] The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM 92] seemed at first as the best possible up to constant factors. A stronger characterization was later conjectured in [BGLR93] one that, as an immediate outcome, implies NP hardness of approximating SET COVER to within logarithmic factors [LY94, BGLR93] The conjecture itself, nonetheless, seems to be of independent interest, allowing a deeper understanding into probabilistic checking of proofs (PCP) and the class NP. ....

[Article contains additional citation context not shown here]

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....L. For most applications of PCP, the characterization of NP with constant error probability and variables of constant range [AS92, ALM 92] suffices. In order to prove NPhardness of other problems, however, sub constant errorprobability has turned out to be essential. For example [LY94] and [BGLR93] were able to prove approximating SET COVER to within logarithmic factors almost NP hard, using the constant error probability PCP characterization of NP. To improve this result to strict NP hardness, BGLR93] had suggested the sliding scale conjecture. The BGLR conjecture states that it is ....

....sub constant errorprobability has turned out to be essential. For example [LY94] and [BGLR93] were able to prove approximating SET COVER to within logarithmic factors almost NP hard, using the constant error probability PCP characterization of NP. To improve this result to strict NP hardness, BGLR93] had suggested the sliding scale conjecture. The BGLR conjecture states that it is possible to keep the number of variables accessed by each local test constant while increasing the variables range, and to achieve error probability polynomially small in the size of the variables range. In ....

[Article contains additional citation context not shown here]

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....any c 1=2 and = 2 Gamma log 1 Gamma1=O(D) n . These parameters give the result claimed above. Notice that our direct reduction immediately implies that a stronger PCP characterization of NP e.g. one with a polynomially small 3 error probability and constant depend as conjectured in [BGLR93] would immediately give NP hardness for approximating Label Cover to within n c for some constant c 0. Structure of the Paper Our main result for Label Cover is proven in section 1. The hardness result for MMSA 3 is proven in section 2, via a reduction from PCP. We then show, in section ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....number of edges to obtain a graph of diameter at most D are NP hard for any fixed D 2: Our reduction shows that both problems BADC and ADC are at least as difficult as the SET COVER or DOMINATING SET problems. From recent non approximability results of Bellare, Goldwasser, Lund, and Russell [2] (see also the survey by Arora [1] follows that unless P=NP there are no constant approximation polynomial time algorithms for both BADC and ADC and any fixed D: After completing a part of this work we learned about the papers [18, 15] Schoone, Bodlaender and van Leeuwen [18] show that ADC is ....

....S 0 that forms a cover and c S : jS 0 j is minimized. Claim. SET COVER 2 is NP hard. We assert that the existence of a polynomial time algorithm for SET COVER 2 will lead to a 2 approximation polynomial algorithm for SET COVER, which is impossible unless P=NP (more precisely, a result in [2, 1] implies that approximating SET COVER within any constant factor is NP hard) Indeed, take an instance S 0 of SET COVER and extend S 0 to the collection S by adding all pairs of elements of X which do not belong to a common set of S 0 : Clearly c S c S 0 : Let S 0 be an optimal solution of SET ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell, Efficient multi--prover interactive proofs with applications to approximation problems, in Proc. 25th ACM Symp. on Theory of Computing, (1993), pp. 113--131.

....as the most natural extension of the old characterization of NP [Coo71] if one has in mind proving hardness results for approximation problems. This characterization has already been used to obtain quite a few hardness results for approximation problems [FGL 91, AS92, ALM 92, PY91, LY94, BGLR93, KLS93, BGS95, Has96, Has96] Previous proofs for PCP characterizations of NP tend to be very complicated, imposing a major obstacle for any attempt of further research in the area. The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM 92] seemed at first as the best ....

....be very complicated, imposing a major obstacle for any attempt of further research in the area. The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM 92] seemed at first as the best possible up to constant factors. A stronger characterization was later conjectured in [BGLR93] one that, as an immediate outcome, implies NP hardness of approximating set cover to within logarithmic factors [LY94, BGLR93] The conjecture itself, nonetheless, seems to be of independent interest, allowing a deeper understanding into probabilistic checking of proofs (PCP) and the class NP. ....

[Article contains additional citation context not shown here]

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....assignment, to within p 2, is NP hard. This is the case since, the outcome of any algorithm that approximates that number to within that factor would determine precisely which of the cases in condition 2 is the one true for the specific input instance. Stronger PCP Characterizations of NP [BGLR93] refined the parameters governing the PCP hierarchy and conjectured a stronger PCP characterization of NP. The possibility of letting variables range over sets containing more than two elements was not considered in [AS92] Once a larger range is allowed, the number of variables each ....

....larger, thereby achieving smaller, hopefully sub constant, error probability. The error probability is defined to be the fraction of functions that can be satisfied by any single assignment in case the input instance is not in the original language. Adopted to the function system terminology, BGLR93] s conjecture is as follows: Conjecture 3 ( BGLR93] For every L 2 NP, let M(n) be at most logarithmic in n, there exists a polynomial time procedure that, on input I 2 f0; 1g n , constructs a set of Booleanfunctions Phi L;I = f 1 ; l g over variables Y = fy 1 ; y l g each ....

[Article contains additional citation context not shown here]

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

....that the geometric facts used in our reduction to SV1might be indicative of the nature of relevant techniques. A related open problem is to improve our results by proving the problems NP hard rather than almost NP hard. More efficient 2 prover, 1 round interactive proofs for NP (as conjectured in [BG ] or a direct reduction from [AL ] might help. Acknowledgements Thanks to Madhu Sudan for suggesting that techniques from interactive proofs might be helpful in proving the hardness of lattice vector problems, and to Ravi Kannan for his prompt responses to questions on lattices. We also thank ....

M. Bellare, S. Goldwasser, C. Lund, A. Russell. Efficient Multi-Prover Interactive Proofs with Applications to Approximation Problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

No context found.

M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient multi-prover interactive proofs with applications to approximation problems. In Proc. 25th ACM Symp. on Theory of Computing, pages 113--131, 1993.

No context found.

M. Bellare, S. Goldwasser, C. Lund, A. Russel, "Efficient Multi-Prover Interactive Proofs with Applications to Approximation Problems", In Proc. 25th ACM Symp. on Theory of Computing, 1993, pp. 113-131.

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