## Improved Non-Approximability Results (1994)

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Citations: | 113 - 15 self |

### BibTeX

@INPROCEEDINGS{Bellare94improvednon-approximability,

author = {Mihir Bellare and Madhu Sudan},

title = {Improved Non-Approximability Results},

booktitle = {},

year = {1994},

pages = {184--193}

}

### Years of Citing Articles

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### Abstract

We indicate strong non-approximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in...

### Citations

502 |
Reducibility Among Combinatorial Problems,” Complexity of Computer Computations
- Karp
- 1972
(Show Context)
Citation Context ...ult is a reduction from Max Clique which is more efficient than previous ones. 1 Introduction Max Clique is amongst the most important combinatorial optimization problems. Unfortunately it is NP-hard =-=[16]-=-, and attention since this discovery has thus focused on approximation algorithms. Yet the best known ones can approximate the max clique size of an N node graph only to within a factor of N 1−o(1) [1... |

405 | Non-Deterministic Exponential Time has TwoProver Interactive Protocols
- Babai, Fortnow, et al.
- 1990
(Show Context)
Citation Context ...mized graph products of [9] to the construction of [12], or apply [25]. The connection described above led researchers to focus on reducing the (expected) number of bits queried in the PCP. Extending =-=[4, 5, 12, 3]-=- it was shown by [2] that N ˜ P is in PCP av [ polylog(n), q ] for a value q which, although not specified in the paper, has been said by the authors to be around 10 4 . Thus they could show that appr... |

390 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1994
(Show Context)
Citation Context ...1/(1+f). Thus we get Theorem 1.1. We can show that that many aspects of our constructions and analysis are tight. We feel that proof systems of a different nature are required to do better. Following =-=[20]-=-, non-approximability results for Chromatic Number have been obtained by “reduction” from the Max Clique problem on the specific set of graphs constructed by the Max Clique reduction of [12]. For the ... |

366 | Probabilistic checking of proofs: A new characterization of NP
- Arora, Safra
- 1992
(Show Context)
Citation Context ...[11], scarcely better than the trivial factor of N. The first twenty years following the NP-hardness discovery brought little understanding of why this is so. Today we know a lot more. The results of =-=[12, 3, 2]-=- indicated the existence of a constant α > 0 for which N α factor approximations are unlikely to be achievable. Recent work [7, 13] has been able to show that α ≥ 1/15. One of the most basic goals in ... |

347 | Self-testing/correcting with applications to numerical problems
- Blum, Luby, et al.
- 1993
(Show Context)
Citation Context ...turn to the proofs of the lemmas. We start with the proof of Lemma 3.6. Proof of Lemma 3.6: Suppose g∗(α) = 0. Let I[y] be the event that g(y+α)−g(y) = 0. Our starting point is the self-corrector of =-=[6, 10]-=- which shows that in this case the probability of I[y] when y is chosen uniformly at random from Zl2 2 is at most 0.2. Observe that for y1, . . . , ym randomly chosen from the space Zl2 2 , every vect... |

269 | Simple construction of almost k-wise independent random variables. Random Struct
- Alon, Goldreich, et al.
- 1992
(Show Context)
Citation Context ...duce this notion and show how the second moment analy-sis can still be applied here. Our notion seems to be slightly weaker than some weak forms of independence that have been used in the literature =-=[21, 1]-=- and incomparable to some of the others [18, 23]. The definition that follows is given only for the special case of boolean random variables, but is easily generalized. We let [N] = {1, . . . , N}. Re... |

256 | Small-bias probability spaces: Efficient constructions and applications
- Naor, Naor
- 1993
(Show Context)
Citation Context ...duce this notion and show how the second moment analy-sis can still be applied here. Our notion seems to be slightly weaker than some weak forms of independence that have been used in the literature =-=[21, 1]-=- and incomparable to some of the others [18, 23]. The definition that follows is given only for the special case of boolean random variables, but is easily generalized. We let [N] = {1, . . . , N}. Re... |

255 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1990
(Show Context)
Citation Context ...mized graph products of [9] to the construction of [12], or apply [25]. The connection described above led researchers to focus on reducing the (expected) number of bits queried in the PCP. Extending =-=[4, 5, 12, 3]-=- it was shown by [2] that N ˜ P is in PCP av [ polylog(n), q ] for a value q which, although not specified in the paper, has been said by the authors to be around 10 4 . Thus they could show that appr... |

205 |
Proof verification and intractability of approximation problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ...[11], scarcely better than the trivial factor of N. The first twenty years following the NP-hardness discovery brought little understanding of why this is so. Today we know a lot more. The results of =-=[12, 3, 2]-=- indicated the existence of a constant α > 0 for which N α factor approximations are unlikely to be achievable. Recent work [7, 13] has been able to show that α ≥ 1/15. One of the most basic goals in ... |

180 |
Approximating clique is almost npcomplete
- Feige, Goldwasser, et al.
- 1991
(Show Context)
Citation Context ...[11], scarcely better than the trivial factor of N. The first twenty years following the NP-hardness discovery brought little understanding of why this is so. Today we know a lot more. The results of =-=[12, 3, 2]-=- indicated the existence of a constant α > 0 for which N α factor approximations are unlikely to be achievable. Recent work [7, 13] has been able to show that α ≥ 1/15. One of the most basic goals in ... |

169 | Efficient probabilistically checkable proofs and applications to approximation
- Bellare, Goldwasser, et al.
- 1993
(Show Context)
Citation Context ...nding of why this is so. Today we know a lot more. The results of [12, 3, 2] indicated the existence of a constant α > 0 for which N α factor approximations are unlikely to be achievable. Recent work =-=[7, 13]-=- has been able to show that α ≥ 1/15. One of the most basic goals in computational complexity theory is to find the exact complexity of prob∗ Advanced Networking Laboratory, IBM T.J. Watson Research C... |

157 |
Hiding instances in multioracle queries
- BEAVER, FEIGENBAUM
- 1990
(Show Context)
Citation Context ...turn to the proofs of the lemmas. We start with the proof of Lemma 3.6. Proof of Lemma 3.6: Suppose g∗(α) = 0. Let I[y] be the event that g(y+α)−g(y) = 0. Our starting point is the self-corrector of =-=[6, 10]-=- which shows that in this case the probability of I[y] when y is chosen uniformly at random from Zl2 2 is at most 0.2. Observe that for y1, . . . , ym randomly chosen from the space Zl2 2 , every vect... |

135 | Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions
- Ben-Or, Goldwasser, et al.
- 1988
(Show Context)
Citation Context ...(1) there is a function δk(n) = o(k(n)) such that L is in FPCP[ (k+δk)r, (k+δk)f, 2 −k , 2 r ]. We’ll also discuss verifiers who talk to a collection of p = p(n) provers rather than to a proof string =-=[8]-=-. Definitions for such things are standard. If Ω is a probability space then ω R ← Ω denotes the operation of selecting an element at random according to Ω, and Pr R [·] is the corresponding probabili... |

135 | Approximating maximum independent sets by excluding subgraps
- Boppana, Hald'orsson
- 1992
(Show Context)
Citation Context ...6], and attention since this discovery has thus focused on approximation algorithms. Yet the best known ones can approximate the max clique size of an N node graph only to within a factor of N 1−o(1) =-=[11]-=-, scarcely better than the trivial factor of N. The first twenty years following the NP-hardness discovery brought little understanding of why this is so. Today we know a lot more. The results of [12,... |

111 |
On the approximation of maximum satisfiability
- Yannakakis
- 1992
(Show Context)
Citation Context .... It is a canonical Max SNP complete problem [22]. An algorithm approximates it within 1 + γ if it produces a value which is at least 1/(1 + γ) times optimal, and at most optimal. An algorithm due to =-=[24]-=- achieves γ = 1/3. Theorem 1.3 Suppose N ˜ P = ˜ P. Then there is no polynomial time algorithm to approximate Max 3SAT within 1 + 1/65. The factor decreases to 1+1/73 if the assumption is just P = N... |

72 | On the Hardness of Approximating the Chromatic Number
- Khanna, Linial, et al.
- 2000
(Show Context)
Citation Context ...for β = 1/(a+bf). Thus the problem is to design (a, b)chromatic number reductions with a, b as small as possible. The reduction of [20] achieves a = 1 and b = 5. (A simple reduction supplied later by =-=[17]-=- is slightly less efficient, achieving a = 6 and b = 5.) Applying this and Theorem 1.4 we can conclude that approximating Chromatic Number within N 1 16 −o(1) is hard, which is already better than the... |

71 |
On the complexity of approximating the independent set problem
- Berman, Schnitger
- 1992
(Show Context)
Citation Context ... α is hard for α = 1/(1+q). This is a slightly tighter version of the original connection of [12] which can be viewed as derived in two alternative ways: either apply the randomized graph products of =-=[9]-=- to the construction of [12], or apply [25]. The connection described above led researchers to focus on reducing the (expected) number of bits queried in the PCP. Extending [4, 5, 12, 3] it was shown ... |

47 |
Two prover protocols - low error at affordable rates
- Feige, Kilian
- 1994
(Show Context)
Citation Context ...ework for proof checking and the idea of reusing query bits. We use both. To reduce below 24 the number of queried bits in a PCP seemed hard. The door to better results was opened by Feige and Kilian =-=[13]-=-. They suggested the notion of free bits (they didn’t name it so; that was our doing) and observed that the Max Clique connection actually achieved what in our language would be the following: if N˜ P... |

34 |
NP-complete problems have a version that’s hard to approximate
- Zuckerman
- 1993
(Show Context)
Citation Context ... and explanations Non-approximability results based on PCPs begin with [12]. They have since been proved under many different assumptions. These include: ˜ P = N ˜ P [12]; P = NP [3, 2]; NP = coRP =-=[25, 7, 13]-=-; and N ˜ P = coR ˜ P [7]. 2 Since our focus is on the factor g(N) shown hard rather than on the assumption, we will talk of a problem being hard to approximate within a certain factor; it is underst... |

32 | Constructing small sample spaces satisfying given constraints
- Koller, Megiddo
- 1994
(Show Context)
Citation Context ... analy-sis can still be applied here. Our notion seems to be slightly weaker than some weak forms of independence that have been used in the literature [21, 1] and incomparable to some of the others =-=[18, 23]-=-. The definition that follows is given only for the special case of boolean random variables, but is easily generalized. We let [N] = {1, . . . , N}. Refer to bottom of Section 2 for notation and conv... |

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

21 | Proof veri and intractability of approximation problems - Arora, Lund, et al. - 1992 |

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

18 | Small-Bias Probability Spaces: Ecient Constructions and Applications - Naor, Naor - 1993 |

16 |
On the second eigenvalue and linear expansion of regular graphs
- Kahale
- 1992
(Show Context)
Citation Context ...the factor shown hard. For example, for any constant δ > 0, assuming NP = coRP we can show that N 1 5 −δ factor approximation is impossible; and assuming just P = NP we can use the constructions of =-=[15]-=- to show that N 1 6 −δ factor approximation is impossible. The technical underpinning of the above result is a new proof system which will be discussed in Section 1.2. We turn now to the chromatic num... |

15 |
Sample spaces uniform on neighborhoods
- SCHULMAN
- 1992
(Show Context)
Citation Context ... analy-sis can still be applied here. Our notion seems to be slightly weaker than some weak forms of independence that have been used in the literature [21, 1] and incomparable to some of the others =-=[18, 23]-=-. The definition that follows is given only for the special case of boolean random variables, but is easily generalized. We let [N] = {1, . . . , N}. Refer to bottom of Section 2 for notation and conv... |

14 | On the Approximation of Maximum Satis - Yannakakis - 1994 |

12 |
Two-prover one round proof systems: Their power and their problems
- Feige, Lovasz
- 1992
(Show Context)
Citation Context ...7], seems to have been forgotten by them; we even reduce —to 13, from the 15 in [7]— the number of 3SAT clauses needed to write the quadratic test. A very skimpy description follows. We first use the =-=[19, 14]-=- proof system to get a canonical two-prover proof system as discussed in Section 3.1. We then convert this into a Max 3SAT by expressing the task of the extended verifier by a 3cnf-formula: A linearit... |

9 |
Fully parallelized multi-prover protocols for NEXP-time
- Lapidot, Shamir
- 1997
(Show Context)
Citation Context ...mation applies to any canonical proof system. The above theorems are obtained by plugging in specific canonical proof systems as we now describe. The only property missing to make the proof system of =-=[19, 14]-=- canonical is that the second prover’s answers are not expressible as a projection of those of the first. This is simple to fix. To see how, recall that the first prover sends a sequence of polynomial... |

8 | Two prover protocols { Low error at aordable rates (preliminary version - Feige, Kilian - 1994 |

2 |
Optimization, approximation, and complexity classes. STOC 88
- Papadimitriou, Yannakakis
(Show Context)
Citation Context ... 14 −δ factor approximation is impossible. Max 3SAT is the problem of determining the maximum number of simultaneously satisfiable clauses in a 3cnfformula. It is a canonical Max SNP complete problem =-=[22]-=-. An algorithm approximates it within 1 + γ if it produces a value which is at least 1/(1 + γ) times optimal, and at most optimal. An algorithm due to [24] achieves γ = 1/3. Theorem 1.3 Suppose N ˜ P ... |