## Efficient probabilistically checkable proofs and applications to approximation (1993)

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Venue: | In Proceedings of STOC93 |

Citations: | 167 - 12 self |

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@INPROCEEDINGS{Bellare93efficientprobabilistically,

author = {M. Bellare and S. Goldwassert and C. Lundi and A. Russell},

title = {Efficient probabilistically checkable proofs and applications to approximation},

booktitle = {In Proceedings of STOC93},

year = {1993},

pages = {294--304}

}

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1

### Citations

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The Theory of Error-Correcting Codes
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- 1977
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Citation Context ... R 2 ; P R 3 and P R 4 ] ? ffl=(2 p ffi ) 4 then x 2 L. Proof: The following lemma implies that each segment of Y 0 i can only be decoded to a constant number of different segments of Y i . Lemma 3.6 =-=[MS]-=- Let A = A(n; d; w) be the maximum number of binary vectors of length n, each with weight (i.e., number of non-zeroes) at most w, and any two of which are distance at least d apart. Then Asdn=(dn \Gam... |

712 |
Approximation Algorithms for Combinatorial Problems
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Citation Context ... x4.1. Recall that there exists a polynomial time algorithm for approximating the size of the minimum set cover to within a factor of \Theta(log N ), where N is the number of elements in the base set =-=[Jo, Lo]-=-. Hardness of approximation was recently shown by Lund and Yannakakis [LY1]. The reduction they used has the property that T is proportional to 2 r+a while Q increases as ffl decreases, and decreases ... |

576 | Optimization, approximation, and complexity classes - Papadimitriou, Yannakakis - 1991 |

406 | Nondeterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...! 1. MIP 1 [r; p; a; q; ffl] denotes the class of languages possessing MIP 1 [r; p; a; q; ffl] verifiers. We let MIP 1 = MIP 1 [poly(n); poly(n); poly(n); poly(n); 1=2]. It is known that MIP 1 = NEXP =-=[BFL]-=-. Probabilistically checkable proofs, as defined in [AS, ALMSS] are the same as what [FRS] had earlier called the oracle model. In this model, a verifier V has ac4 cess to an input x 2 f0; 1g n and an... |

394 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1992
(Show Context)
Citation Context ...n optimal solution to an instance of problem P . Today such results are known for many important problems P , with values of Q and T which differ from problem to problem; for example, it was shown by =-=[LY1]-=- that approximating the size of the minimum set cover to within \Theta(log N ) implies NP ` DTIME(n polylog n ), and it was shown by [FGLSS, AS, ALMSS] that for some constant c ? 0 approximating the s... |

277 | On the ratio of optimal integral and fractional cover - Lovasz - 1975 |

257 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...cing Randomness As discussed above, the transformation of [FL, LS] reduces the error without increasing the number of provers, but costs in randomness. We combine this transformation with the idea of =-=[BFLS] of using -=-as "base field" not f0; 1g but some larger subset H of the underlying finite field F . Choosing h = log jHj appropriately we need use only O(k(n) log n) random bits to get error 2 \Gammak(n)... |

204 |
Proof Verification and Intractability of Approximation Problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ... proof systems in question and what within these proof systems are the complexity parameters we need to consider. r = r(n) p = p(n) a = a(n) ffl = ffl(n) How (in a word) (1) O(log n) 2 O(1) \Theta(1) =-=[ALMSS]-=-+[FRS] (2) O(log n) O(k(n)) O(1) 2 \Gammak(n) O(k(n)) [CW, IZ]-style repetitions of (1). (3) O(k(n) log 2 n) 2 O(k(n) log 2 n) 2 \Gammak(n) [FL] (4) O(k(n) log n) + poly(k(n); log log n) 4 poly(k(n); ... |

187 | How to Recycle Random Bits - Impagliazzo, Zuckerman - 1989 |

179 | Approximating clique is almost NP-complete - Feige, Goldwasser, et al. - 1991 |

135 | Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions
- Ben-Or, Goldwasser, et al.
- 1988
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Citation Context ...ave been used to derive intractability of approximation results. We focus on two of them. The first is the (single round version of the) multi-prover model of Ben-Or, Goldwasser, Kilian and Wigderson =-=[BGKW]. The second is the -=-"oracle" model of Fortnow, Rompel and Sipser [FRS], renamed "probabilistically checkable proofs" by Arora and Safra [AS]. In each case, we may distinguish five parameters which we ... |

131 | On the Power of Multi-Prover Interactive Protocols
- Fortnow, Rompel, et al.
- 1988
(Show Context)
Citation Context ...ystems in question and what within these proof systems are the complexity parameters we need to consider. r = r(n) p = p(n) a = a(n) ffl = ffl(n) How (in a word) (1) O(log n) 2 O(1) \Theta(1) [ALMSS]+=-=[FRS]-=- (2) O(log n) O(k(n)) O(1) 2 \Gammak(n) O(k(n)) [CW, IZ]-style repetitions of (1). (3) O(k(n) log 2 n) 2 O(k(n) log 2 n) 2 \Gammak(n) [FL] (4) O(k(n) log n) + poly(k(n); log log n) 4 poly(k(n); log lo... |

90 | Dispersers, deterministic amplification, and weak random sources - Cohen, Wigderson - 1989 |

76 | Structure preserving reductions among convex optimization problems - Ausiello, D’Atri, et al. - 1980 |

61 | The Theory of Error–Correcting Codes - Williams, Sloane - 1978 |

59 | The complexity of approximating a nonlinear program - Bellare, Rogaway - 1995 |

53 | The approximation of maximum subgraph problems - Lund, Yannakakis - 1993 |

49 | Approximation properties of NP minimization classes - Kolaitis, Thakur - 1995 |

34 |
NP-complete problems have a version that’s hard to approximate
- Zuckerman
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Citation Context ... c is NP-complete; this result uses randomness efficient error reduction techniques such as [CW, IZ, BGG], and c depends on t; d as well as other constants arising from the error-reduction. Zuckerman =-=[Zu]-=- uses a random construction which achieves c = 1=t at the cost of weakening the conclusion to NP ` BPP. Based on Theorem 1.2, we can improve the value of c in this result. 2 This is a simplification o... |

27 |
Approximating clique is NP-complete
- Arora, Safra
- 1992
(Show Context)
Citation Context ...-prover model of Ben-Or, Goldwasser, Kilian and Wigderson [BGKW]. The second is the "oracle" model of Fortnow, Rompel and Sipser [FRS], renamed "probabilistically checkable proofs"=-= by Arora and Safra [AS]-=-. In each case, we may distinguish five parameters which we denote by r; p; a; q and ffl (all are in general functions of the input length n). In a multi prover proof these are, respectively, the numb... |

20 |
R.: Selftesting and Self-correcting programs, with Applications to Numerical Programs
- Blum, Luby, et al.
- 1990
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Citation Context ...n step (Lemma 3.2 and Lemmas 3.5 and 3.10) which allows us to combine proof systems with almost no increase in the error probability. Furthermore we have an improved analysis of the linearity test in =-=[BLR]-=- and of the matrix multiplication test for the special case needed in [ALMSS]. We will now sketch our construction. Some knowledge of the proof in [ALMSS] is assumed of the reader. First we need some ... |

20 | The complexity of the max-word problem and the power of one-way interactive proof systems - Condon - 1993 |

19 | Interactive proofs and approximation: reductions from two provers in one round - Bellare - 1993 |

12 |
Two-prover one round proof systems: Their power and their problems
- Feige, Lovasz
- 1992
(Show Context)
Citation Context ...ffl(n) How (in a word) (1) O(log n) 2 O(1) \Theta(1) [ALMSS]+[FRS] (2) O(log n) O(k(n)) O(1) 2 \Gammak(n) O(k(n)) [CW, IZ]-style repetitions of (1). (3) O(k(n) log 2 n) 2 O(k(n) log 2 n) 2 \Gammak(n) =-=[FL]-=- (4) O(k(n) log n) + poly(k(n); log log n) 4 poly(k(n); log log n) 2 \Gammak(n) This paper. Figure 1: Number of random bits (r), number of provers (p), answer size (a) and error probability (ffl) in r... |

9 |
Fully parallelized multi-prover protocols for NEXP-time
- Lapidot, Shamir
- 1997
(Show Context)
Citation Context ...nd to prover i. Regard each Q i as divided into s pieces, each of size h (that is, Q i = Q 1 i : : : Q s i with each Q j i in H) so that Q i is an element of H s ` F s . We now apply the technique of =-=[LS]-=-. V 0 chooses, randomly and uniformly, y 1 ; : : : ; y p from F s . For each i we let l i : F ! F s be (a canonical representation of) the unique line through Q i and y i . Let t i;1 ; t i;2 2 F satis... |

8 | Proof veri cation and intractability of approximation Problems - Arora, Lund, et al. - 1998 |

7 | Interactive Proofs and Approximation - Bellare - 1992 |

7 |
PCP and tighter bounds for approximating MAXSNP
- PHILLIPS, SAFRA
- 1992
(Show Context)
Citation Context ...oximation. The (constant) number of bits t that one needs to check in the [ALMSS] result NP ` PCP[O(logn); t; 1; O(logn); 1=2] is of the order of 10 4 . Some reductions in this value were obtained by =-=[PS]-=-. Our improved complexity of four prover proofs for NP in terms of randomness and answer size for a given error, together with a careful analysis of the [ALMSS] construction and proofs enable us to ob... |

5 | Highly resilient correctors for polynomials. Information processing letters - Gemmel, Sudan - 1992 |

4 | On the Success Probability of the Two - Feige - 1991 |

4 | Dispersers, Deterministic Ampli cation, and Weak Random Sources - Cohen, Wigderson - 1989 |

3 | On the ratio of optimal integral and fractional covers - asz - 1975 |

3 |
On approximation algorithms for concave programming
- Vavasis
- 1992
(Show Context)
Citation Context ...m of f over the feasible region, and by fsthe minimum. Following [ADP, Va] we say that ~ f is a ��-approximation, where �� 2 [0; 1], if jf \Gamma ~ f js�� \Delta jf \Gamma fsj. We refer th=-=e reader to [Va]-=- for discussion of the appropriateness of this definition, and the inappropriateness of the one we have been using above, in the context of continuous optimization problems. Quadratic programming has ... |

2 | Optimization, approximation, and complexity classes. STOC 88 - Papadimitriou, Yannakakis |

2 | On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383390 - LOVSZ - 1975 |

1 | Approximation properties of NP minimization classes. Strut ture in Complexity Theory - KOLAITI, THAKUR - 1991 |