## Proof verification and hardness of approximation problems (1992)

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Venue: | IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI |

Citations: | 717 - 46 self |

### BibTeX

@INPROCEEDINGS{Arora92proofverification,

author = {Sanjeev Arora and Carsten Lund and Rajeev Motwani and Madhu Sudan and Mario Szegedy},

title = {Proof verification and hardness of approximation problems},

booktitle = {IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI},

year = {1992},

pages = {14--23},

publisher = {}

}

### Years of Citing Articles

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### Abstract

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 SNP-hard 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 N-vertex graph to within a factor of N ɛ is NP-hard.

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Citation Context ... if x and x ′ are two different strings of length n then the hamming distance dist(x, x ′ ) between E(x) and E(x ′ ) is at least δn. An encoding scheme can be constructed using the Reed-Solomon codes =-=[MS81]-=-. For an encoding scheme we define the decoding E −1 (z) as the x that minimizes dist(E(x), z). 5.2 Circuit verification The theorem of proof verification in [BFLS91] we turn into circuit verification... |

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Citation Context ...alesman problem with triangle inequality, minimal steiner tree, maximum directed cut, shortest superstring, etc. be-long to the class of MAXSNP-hard problems, defined by Papadimitriou and Yannakakis =-=[PY91]-=- in terms of logic and reductions that preserves approximability. Our result also improves the parameters for the MAX-CLIQUE result of Arora and Safra. We show, that there is an ɛ > 0 such that approx... |

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Citation Context ...d to surprisingly strong hardness results for approximating optimization problems. Feige et al. exploited a recent characterization of multiprover interactive proof systems by Babai, Fortnow and Lund =-=[BFL91]-=- to obtain intractability results for approximating MAX-CLIQUE under the assumption that NP ̸⊆ DT IME(n O(log log n) ). Recently Arora and Safra [AS92] improved on this by showing that it is NP-hard t... |

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Citation Context ...r example, MAX SAT can only be approximated to a ratio of 4/3 [Yan92], while MAX CUT and vertex cover can only be approximated to a ratio of 2 [GJ79, Mot92]. The recent results of Lund and Yannakakis =-=[LY92]-=- showed that the chromatic number is as hard to approximate as the clique and thus solce a long-standing open problem. They also show that the logarithmic ratio achievable for the set cover problem is... |

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Citation Context ...active proof systems by Babai, Fortnow and Lund [BFL91] to obtain intractability results for approximating MAX-CLIQUE under the assumption that NP ̸⊆ DT IME(n O(log log n) ). Recently Arora and Safra =-=[AS92]-=- improved on this by showing that it is NP-hard to approximate MAXCLIQUE within any constant factor (and even within log n/(log log n)O(1) a factor of 2 ). Their solution builds on and further develop... |

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Citation Context ...to approximate MAX-CLIQUE within a factor of n ɛ is NP-hard. 1.3 Related Areas The results in this paper borrow significantly from results in the area of self-testing/self-correcting of programs (see =-=[BLR90]-=-, [Rub90]). The areas of selftesting/correcting are closely connected to the areas of error-detection/correction in coding theory. In particular, we observe that results from the former area can be in... |

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Citation Context ...he above definition of NP leads to definitions of interesting new complexity classes, which have been the subject of intense research in the past decade. Goldwasser, Micali and Rackoff [59] and Babai =-=[10, 16]-=- allowed the verifier to be a probabilistic polynomialtime Turing Machine that interacts with a “prover,” which is an infinitely powerful Turing Machine trying to convince the verifier that the input ... |

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Citation Context ...imal solutions. The task of proving hardness of the approximation versions of such problems met with limited success. For the traveling salesman problem without triangle inequality Sahni and Gonzalez =-=[SG76]-=- showed ∗ Computer Science Division, U. C. Berkeley, Berkeley, CA 94720. Supported by NSF PYI Grant CCR 8896202. † AT&T Bell Labs, Murray Hill, NJ 07974. ‡ Department of Computer Science, Stanford Uni... |

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Citation Context ...to achieve worst-case performance ratio α(n) if for every input x : F (x, A(x)) ≥ α(n) −1optF (x), where n is the size of x. In 1988 Papadimitriou and Yannakakis [PY91] using Fagin’s definition of NP =-=[Fag74]-=- observed that there is an approximation algorithm which has constant performance ratio for any maximization problem that is defined by a quantifier free first order formula ϕ as optϕ(X) = max : |{z|ϕ... |

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Citation Context ...by Arora and Safra [AS92], as a slight variation of the notions of randomized oracle machines due to Fortnow, Rompel and Sipser [FRS88] and transparent proofs due to Babai, Fortnow, Levin and Szegedy =-=[BFLS91]-=-. All these models are in turn variations of interactive proof systems [Bab85, GMR89] and multiprover interactive proof systems [BGKW88]. Definition 1.3 (Arora-Safra [AS92]) A language L is in PCP(f(n... |

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185 |
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Citation Context ...st part, not much could be said for a wide variety of problems until very recently. A connection between two seemingly unrelated areas within theoretical computer science, established by Feige et al. =-=[FGLSS91]-=-, led to surprisingly strong hardness results for approximating optimization problems. Feige et al. exploited a recent characterization of multiprover interactive proof systems by Babai, Fortnow and L... |

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Citation Context ...t some specified point x ∈ F m . We now describe a procedure which computes p(x) using few probes into O and an auxiliary oracle B. The procedure owes its origins to the work of Beaver and Feigenbaum =-=[17]-=- and Lipton [76]. The specific analysis given below is borrowed from the work of Gemmell, Lipton, Rubinfeld, Sudan and Wigderson [56] and allows the number of queries to be independent of d, for error... |

155 | Interactive proofs and the hardness of approximating cliques - Feige, Goldwasser, et al. - 1996 |

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Citation Context ...ertex deletion.) Khanna, Linial and Safra [71] study the hardness of coloring 3-colorable graph. They show that coloring a 3-colorable graph with 4 colors is NP-hard. Arora, Babai, Stern, and Sweedyk =-=[3]-=- prove hardness results for a collection of problems involving integral lattices, codes, or linear equations/inequations. These include Nearest Lattice Vector, Nearest Codeword, and the Shortest Latti... |

146 | Improved low-degree testing and its applications
- Arora, Sudan
(Show Context)
Citation Context ...erifiers making constant number of queries with logarithmic randomness and answer size, where the error is as low as 2− log1−ɛ n for every ɛ > 0. An alternate construction is given in Arora and Sudan =-=[7]-=-. Better non-approximability results. Part of the motivation for improving the construction of outer verifiers is to improve the ensuing non-approximability results. The result for MAX-3SAT in this pa... |

143 | The Traveling Salesman Problem With Distances One and Two - Papadimitriou, Yannakakis - 1993 |

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(Show Context)
Citation Context ...and transparent proofs due to Babai, Fortnow, Levin and Szegedy [BFLS91]. All these models are in turn variations of interactive proof systems [Bab85, GMR89] and multiprover interactive proof systems =-=[BGKW88]-=-. Definition 1.3 (Arora-Safra [AS92]) A language L is in PCP(f(n), g(n)) if there is polynomial-time randomized oracle machine M y (r, x) which works as follows: 1. It takes input x and a (random) str... |

129 | On the power of multi-prover interactive protocols
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(Show Context)
Citation Context ...s The notion of probabilistically checkable proofs (PCP) was introduced by Arora and Safra [AS92], as a slight variation of the notions of randomized oracle machines due to Fortnow, Rompel and Sipser =-=[FRS88]-=- and transparent proofs due to Babai, Fortnow, Levin and Szegedy [BFLS91]. All these models are in turn variations of interactive proof systems [Bab85, GMR89] and multiprover interactive proof systems... |

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108 |
On the approximation of maximum satisfiability
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(Show Context)
Citation Context ...ap between the (negligible) constants in the hardness results and the approximation ratio currently achievable for the MAXSNP problems. For example, MAX SAT can only be approximated to a ratio of 4/3 =-=[Yan92]-=-, while MAX CUT and vertex cover can only be approximated to a ratio of 2 [GJ79, Mot92]. The recent results of Lund and Yannakakis [LY92] showed that the chromatic number is as hard to approximate as ... |

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(Show Context)
Citation Context ...cy due to a technical lemma from [AS92], as an efficient mechanism to test Reed Solomon Codes. Other ingredients in our proof borrow from work done in “parallelizing” the MIP=NEXPTIME protocol [LS91],=-=[FL92]-=-. The result described in Section 7 uses ideas from their work. 2 PCP and MAXSNP The methods of [FGLSS91] and [AS92] have been applied so far only to the clique approximation problem. Here we show tha... |

94 |
Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems
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(Show Context)
Citation Context ...significant progress on designing better approximation algorithms for some of the problems mentioned earlier. Two striking results in this direction are those of Goemans and Williamson [57] and Arora =-=[2]-=-. Goemans and Williamson [57] show how to use semidefinite programming to give better approximation algorithms for MAX-2SAT and MAX-CUT. Arora [2] has discovered a polynomial time approximation scheme... |

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Citation Context ...+ yQ = yP +Q) ≥ 0.99. The existence of L now follows directly from the following lemma, due to Blum, Luby and Rubinfeld [BLR90]. The bound we state here appears in Rubinfeld [Rub90] and Gemmel et al. =-=[GLRSW91]-=-. Lemma 6 Let g be a function such that P robx,y(g(x) + g(y) ̸= g(x + y)) ≤ δ/2 then there exists a linear function L such that P rob(g(x) ̸= L(x)) ≤ δ. Whereas yP is not always L(P ), Equation 5 give... |