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## 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: | 792 - 38 self |

### Citations

13850 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

2646 |
The theory of error-correcting codes
- MacWilliams, Sloane
- 1977
(Show Context)
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... |

1936 | Reducibility among combinatorial problems - Karp - 1972 |

1234 | The knowledge complexity of interactive proof systems - Goldwasser, Micali, et al. - 1989 |

1195 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming - Goemans, Williamson - 1995 |

1033 | The complexity of theoremproving procedures - Cook - 1971 |

821 | Approximation algorithms for combinatorial problems - Johnson - 1974 |

748 | Some optimal inapproximability results - Håstad - 2001 |

606 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
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... |

468 | Some simplified NP-complete graph problems - Garey, Johnson, et al. - 1976 |

424 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1994
(Show Context)
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... |

417 | Nondeterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
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... |

413 | Probabilistic checking of proofs: A new characterization of NP
- Arora, Safra
- 1998
(Show Context)
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... |

382 | Bounds for certain multiprocessing anomalies - Graham - 1966 |

365 | Self-testing/correcting with applications to numerical problems
- Blum, Luby, et al.
- 1993
(Show Context)
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... |

365 | Robust characterization of polynomials with applications to program testing - Rubinfeld, Sudan - 1996 |

364 | P-complete approximation problems
- Sahni, Gonzalez
- 1976
(Show Context)
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... |

360 | A parallel repetition theorem - Raz - 1998 |

351 | Trading group theory for randomness
- Babai
- 1985
(Show Context)
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 ... |

350 | Designing programs that check their work - Blum, Kannan - 1989 |

338 | Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes - Babai, Shlomo - 1988 |

336 | Algebraic methods for interactive proof systems - Lund, Fortnow, et al. - 1992 |

328 | A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP - Raz, Safra - 1997 |

313 |
Generalized first-order spectra and polynomialtime recognizable sets
- Fagin
- 1974
(Show Context)
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... |

262 | Checking Computations in Polylogarithmic Time
- Babai, Fortnow, et al.
- 1991
(Show Context)
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... |

212 | Free bits, PCPs, and nonapproximability - towards tight results - Bellare, Goldreich, et al. - 1998 |

196 | How to recycle random bits - Impagliazzo, Zuckerman - 1989 |

194 | Zero Knowledge and the Chromatic number - Feige, Kilian - 1998 |

184 | Guillotine subdivisions approximate polygonal subdivisions: part II — a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems - Mitchell - 1999 |

173 |
Approximating Clique is almost NP-complete
- Feige, Goldwasser, et al.
(Show Context)
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... |

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

166 | The hardness of approximate optima in lattices, codes, and systems of linear equations
- Arora, Babai, et al.
- 1997
(Show Context)
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... |

162 | Efficient probabilistically checkable proofs and applications to approximation - BELLARE, GOLDWASSER, et al. - 1993 |

161 |
Hiding instances in multioracle queries
- Beaver, Feigenbaum
- 1990
(Show Context)
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... |

160 | Clique is hard to approximate within n 1−ɛ - Hastad - 1999 |

148 | On the power of multi-prover interactive protocols
- Fortnow, Rompel, et al.
- 1988
(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... |

148 | The travelling salesman problem with distances one and two - Papadimitriou, Yannakakis - 1993 |

147 | Multi-Prover Interactive Proofs: How to Remove Intractability
- Ben-Or, Goldwasser, et al.
- 1988
(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... |

144 | Improved low-degree testing and its applications
- Arora, Sudan
- 1997
(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... |

139 | An e cient approximation scheme for the one-dimensional bin-packin g problem - Karmaker, Karp - 1982 |

132 | Error correction for algebraic block codes - Welsh, Berlekamp - 1986 |

123 | On syntactic versus computational views of approximability - Khanna, Motwani, et al. - 1998 |

113 | Strong NP-Completeness Results: Motivation, Examples and Implications - Garey, Johnson - 1978 |

111 |
On the approximation of maximum satisfiability
- Yannakakis
- 1994
(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 ... |

110 | Improved nonapproximability results - BELLARE, SUDAN - 1994 |

107 | Two-prover one-round proof systems: Their power and their problems
- Feige, Lovász
- 1992
(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... |

102 | New directions in testing - Lipton - 1991 |

101 |
Polynomial-time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1996
(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... |

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

85 | Approximate algorithms for the 0/1 knapsack problem - Sahni - 1975 |

81 | The Steiner problem with edge lengths 1 and 2 - BERN, PLASSMANN - 1989 |

80 | Structure Preserving Reductions among Convex Optimization Problems - Ausiello, D'Atri, et al. - 1980 |

78 | hardness of approximating the chromatic number - Khanna, Motwani, et al. - 1993 |

76 | Linear approximation of shortest superstrings - Blum, Jiang, et al. - 1994 |

76 | Probabilistic Checking of Proofs and Hardness of Approximation Problems - Arora - 1994 |

75 | Self-testing/correcting for polynomials and for approximate functions
- Gemmell, Lipton, et al.
- 1991
(Show Context)
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... |

74 | Maximum bounded 3-dimensional matching is max snp-complete - Kann - 1991 |

71 |
On the complexity of approximating the independent set problem
- BERMAN, SCHNITGER
- 1992
(Show Context)
Citation Context ...ph products to construct “gap increasing reductions” that mapped an instance of the clique problem into another instance in order to enlarge the relative gap of the clique sizes. Berman and Schnitger =-=[BS92]-=- used the ideas of increasing gaps to show that if the clique size of graphs with bounded co-degree (degree of the complement) does not have a randomized PTAS then there is an ɛ > 0 such that MAX-CLIQ... |

70 |
The complexity of near-optimal graph coloring
- Garey, Johnson
- 1976
(Show Context)
Citation Context ..., Berkeley, CA 94720. Supported by NSF PYI Grant CCR 8896202. ¶ AT&T Bell Labs, Murray Hill, NJ 07974. that finding a solution within any constant factor of optimal is also NP-hard. Garey and Johnson =-=[GJ76]-=- studied MAX-CLIQUE: the problem of finding the largest clique in a graph. They showed that if a polynomial time algorithm computes MAXCLIQUE within a constant multiplicative factor then MAX-CLIQUE ha... |

68 | The complexity of multiway cuts - Dahlhaus, Johnson, et al. - 1992 |

68 | On approximating the longest path in a graph - Karger, Motwani, et al. - 1997 |

66 | Linearity testing in characteristic two - Bellare, Coppersmith, et al. |

66 | Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems - Sudan - 1996 |

65 |
Fast Probabilistic Algorithms
- Freivalds
- 1979
(Show Context)
Citation Context ...) = vT Zw. If the linear forms L1 and L2 are drawn randomly and independently from P1 then the vectors v and w are drawn randomly and independently from GF (2) l . We now use the argument of Freivald =-=[Fre79]-=- to show that Y = Z. It is easily verified that if Y ̸= Z then with probability at least 0.5, v T Y ̸= v T Z. Moreover, for vectors α, β ∈ Z n 2 and w∈RZ n 2 , if α ̸= β then with probability at least... |

63 |
Hardness of approximations. In Approximation Algorithms for NP-hard problems
- Arora, Lund
- 1996
(Show Context)
Citation Context ... hardness of approximating the longest path in a graph to within a 2log1−ɛ n factor, for any ɛ > 0. There are many other results which we haven’t mentioned here; see the compendium [37] or the survey =-=[4]-=-. Improved analysis of outer verifiers. Our construction of an efficient outer verifier for NP languages (Theorem 17) can be viewed as constructing a constant prover 1-round proof system that uses O(l... |

50 | Highly resilient correctors for polynomials - Gemmell, Sudan - 1992 |

48 | Arithmetization: a new method in structural complexity theory - Babai, Fortnow - 1991 |

48 | Two prover protocols - low error at affordable rates - Feige, Kilian - 1994 |

42 | Non deterministic polynomial optimization problems and their approximations. Theoret - Paz, Moran - 1981 |

41 | A survey of Russian approaches to Perebor (brute-force search) algorithms - Trakhtenbrot - 1984 |

40 |
A Mathematical Theory of Self-Checking, Self-Testing, and SelfCorrecting Programs
- Rubinfeld
- 1990
(Show Context)
Citation Context ...imate 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 interpreted... |

38 | Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions - Condon, Feigenbaum, et al. - 1993 |

37 | Universal’nyĭe perebornyĭe zadachi (Universal search problems - Levin - 1973 |

37 |
Fully parallelized multi prover protocols for NEXPTIME
- Lapidot, Shamir
- 1991
(Show Context)
Citation Context ...fficiency 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 s... |

37 | The decision problem for the probabilities of higher-order properties - Kolaitis, Vardi - 1987 |

36 | Random debaters and the hardness of approximating stochastic functions - Condon, Feigenbaum, et al. - 1997 |

35 | Testing of the long code and hardness for clique - Håstad - 1996 |

33 |
Testing polynomial functions efficiently and over rational domains
- Rubinfeld, Sudan
- 1992
(Show Context)
Citation Context ... and Rubinfeld [BLR90] as an efficient error-detection scheme for the Hadamard Codes and this plays a crucial role in our proof. Later in Section 7 we use the “low-degree test” of Rubinfeld and Sudan =-=[RS92]-=-, with its improved efficiency 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 M... |

27 | Lecture Notes on Approximation Algorithms - Motwani - 1992 |

26 | Selftesting /correcting with applications to numerical problems - Blum, Luby, et al. - 1990 |

24 | On approximating the longest path in a graph. Algorithmica - Karger, Motwani, et al. - 1997 |

22 | The complexity of the max word problem, or the power of one-way interactive proof systems - Condon - 1991 |

21 | Ecient probabilistically checkable proofs - Bellare, Goldwasser, et al. - 1993 |

20 | Low-degree testing - HÁTSÁGI, SHEN - 1991 |

19 | Bin packing can be solved within 1 + ɛ in linear time - Vega, Lueker - 1981 |

18 | Interactive proofs and approximation: reduction from two provers in one round - Bellare - 1993 |

16 | Toward a unified approach for the classification of NP-complete optimization problem - Ausiello, Marchetti-Spaccamela, et al. - 1980 |

13 | A Taxonomy of Proof Systems - Goldreich - 1997 |

9 | Some improvements to low-degree tests - Friedl, Sudan - 1995 |

8 |
Transparent (holographic) proofs
- Babai
- 1993
(Show Context)
Citation Context ...r proof of the main theorem, we showed how to encode an assignment to a given circuit so that somebody else can check that it is a satisfying assignment by looking at O(1) bits in the encoding. Babai =-=[11]-=- raised the following question: How efficient can this encoding be? In our paper, encoding an assignment of size n requires poly(n) bits. This was reduced to n 2+ɛ by Sudan [95]. The main hurdle in fu... |

8 | Nearly Linear Size Holographic Proofs - POLISHCHUK, SPIELMAN - 1994 |

6 | On slightly superlinear transparent proofs - Babai, Friedl - 1993 |

6 |
NP-completeness results: Motivation, examples, and implications
- “‘Strong’
- 1978
(Show Context)
Citation Context ...largest clique in a graph. They showed that if a polynomial time algorithm computes MAXCLIQUE within a constant multiplicative factor then MAX-CLIQUE has a polynomial-time approximation scheme (PTAS) =-=[GJ78]-=-, i.e., for any c > 1 there exists a polynomial-time algorithm that approximates MAX-CLIQUE within a factor of c. They used graph products to construct “gap increasing reductions” that mapped an insta... |

5 | Complexity in numerical optimization - Pardalos - 1993 |

3 | A threshold of ln n for Set Cover - Feige - 1996 |

3 | Multi-prover encoding schemes and three prover proof systems - Tardos - 1996 |

2 |
PCP and approximation problems. Unpublished note
- Arora, Motwani, et al.
- 1992
(Show Context)
Citation Context ...f checking and non-approximability (Bellare [18], Bellare and Rogaway [22], Feige and Lovász[45], and Zuckerman [100]). Another such connection is reported by Arora, Motwani, Safra, Sudan and Szegedy =-=[5]-=-, which shows the connection between PCP’s and the hardness of approximating MAX 3SAT. The following theorem summarizes this result; for a proof see Section 3. Theorem 3 ([5]) If NP ⊂ ∪c>0PCP(c log n,... |

2 | Towards a Uni Approach for the Classi of NP-complete Optimization Problems. Theoretical Computer Science - Ausiello, Marchetti-Spaccamela, et al. - 1980 |

1 | Dipartimento di Scienze dell’Informazione, Università di Roma “La Sapienza”, SI/RR-95/02 - Report - 1995 |

1 | A corrected English translation appears - Informatsii - 1973 |