## Interactive proofs and the hardness of approximating cliques (1996)

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Venue: | Journal of the ACM |

Citations: | 151 - 10 self |

### BibTeX

@ARTICLE{Feige96interactiveproofs,

author = {Uriel Feige and Shafi Goldwasser and Laszlo Lovasz and Shmuel Safra and Mario Szegedy},

title = {Interactive proofs and the hardness of approximating cliques},

journal = {Journal of the ACM},

year = {1996},

volume = {43},

pages = {268--292}

}

### Years of Citing Articles

### OpenURL

### Abstract

The contribution of this paper is two-fold. First, a connection is shown between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multiprover interactive proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions. 1

### Citations

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(Show Context)
Citation Context ...listic Oracle Machines An alternative proof of the second part of Theorem 4 is to take the graph Gx = 〈V, E〉 which corresponds to running the protocol once and define the following graph product (see =-=[23]-=-): G 2 x = 〈V ′ , E ′ 〉 where V ′ = V ×V and (〈v1, v2〉 , 〈v ′ 1, v ′ 2〉) ∈ E ′ iff {(v1, v ′ 1) ∈ E or v1 = v ′ 1} and {(v2, v ′ 2) ∈ E or v2 = v ′ 2}. It is easy to see that ω(G 2 x) = ω(Gx) 2 , and ... |

1430 |
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Citation Context ...two of which are connected by an edge. Computing the size of the largest clique in a graph G, denoted ω(G), is one of the first problems shown NP-complete in Karp’s well known paper on NPcompleteness =-=[26]-=-. In this paper, we consider the problem of approximating ω(G) within a given factor. We say that function f(x) approximates (from below) g(x) ≤ h(x). within a factor h(x) iff 1 ≤ g(x) f(x) ∗ Departme... |

706 | Proof verification and the hardness of approximation problems
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Citation Context ...nd Safra have shown that in fact, NP ⊂ P CP (log n, √ log n). They concluded that ω(G) cannot be approximated within a factor of 2 O( less P = NP. √ log n), unArora, Lund, Motwani, Sudan, and Szegedy =-=[2]-=- improved upon the work of [3], and showed that NP ⊂ P CP (log n, 1). This implied that it is NP-hard to approximate ω(G) within a factor of n ɛ , for some ɛ > 0. More importantly, the quantitative im... |

567 |
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Citation Context ...nimum size vertex cover can be approximated within a factor of 2 − Ω( log n ) [6, 30] in polynomial time. No NP-hardness results was known for approximating vertex cover. Papadimitriou and Yannakakis =-=[32]-=- initiated a classification of NP optimization problems based on their logical characterization. They define the class MAX NP, and use its logical characterization to infer that all problems in this c... |

394 | Non-Deterministic Exponential Time has Two-Prover Interactive Protocols
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(Show Context)
Citation Context ...tion of [19]) in which the number of random bits used by the verifier and answer bits sent by the oracle is small — O(log n · log log n). To do so, we consider the theorem of Babai, Fortnow and Lund (=-=[5]-=-), showing that NEXPTIME has multi-prover interactive proofs (This theorem implies that approximating the acceptance probability of multi-prover interactive proof systems on a given input is NEXPTIME-... |

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Citation Context ...at of vertex cover, mentioned in Section 1.3. Both our reduction from PCP protocols to clique, and the [2] reduction to MAX3SAT, treat the PCP characterization of NP as a blackbox. Lund and Yannakakis=-=[29]-=- looked more carefully at the structure of the protocols that give this characterization. Using this structure, they derived sophisticated reductions that showed that it is NP-hard to approximate the ... |

359 | Probabilistic checking of proofs: A new characterization of NP
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Citation Context ...ive proofs (in their various forms) and hardness of approximation problems. Both of the open problems we posed received affirmative answers. The first of these questions was solved by Arora and Safra =-=[3]-=-. To describe their results we adopt their notation, which has become standard by now. Let P CP (r(n), c(n)) denote the class of languages that have probabilistically checkable proofs in which the ver... |

302 | Algebraic Methods for Interactive Proof Systems
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Citation Context ...s, 1 the use of pseudo-random sampling, and a tighter analysis of the multilinearity test. We now proceed to describe the ingredients of our protocol. 4.1 Arithmetization We follow ideas developed in =-=[28]-=-, [33], [5]. 1 A similar change is also suggested in [5] and [4]. 12sIf suffices to show a procedure for deciding the NP-complete language 3-SAT. Let f be a given 3CNF formula of length n. Let m be th... |

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Citation Context ...andomness and answer size (which is sufficient to obtain the result that ω(G) is hard to approximate within a factor of 2 log1−ɛ n for any ɛ > 0, unless NP ⊆ ˜P), was obtained [16] independently from =-=[4]-=- who also scaled down [5]’s protocol to the NP level. In their work, they bound the total running time of the verifier by poly(logarithmic) time rather than the particular parameters of randomness and... |

188 | The NP-completeness column: an ongoing guide
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Citation Context ... that set cover cannot be approximated within a factor of (log n)/4 unless ˜P = N˜P. (Approximating set cover within a factor of ln n is in P.) For further references, see the survey paper of Johnson =-=[25]-=-, and the bibliographical list in [15]. 1.5 Roadmap The paper is organized as follows: In section 2 we introduce some notation and the model of multi-prover interactive proofs. In section 3 we show th... |

185 |
Approximating clique is almost NP-complete
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- 1991
(Show Context)
Citation Context ...hat approximates the size n of the maximum clique in an n-node graph up to a factor of logO(1) n . 1.4 Subsequent Work on Approximation When our work first appeared, we raised two major open problems =-=[17]-=-. 1. Can the methods be extended to prove that if the clique function can be approximated to within factor f, then P = NP? Say, even for f = 2? Using a similar proof outline to the one we use here to ... |

159 |
Proofs that yield nothing but their validity and a method of cryptographic protocol design
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Citation Context ... both works are similar. Our work shows an interesting relation between work on interactive proof systems and long standing open problems in computational complexity theory. For earlier examples, see =-=[21]-=-, [11], [18], [14], [13]. 3s1.2 The Multi Linearity Test An important ingrediant in the [5] protocol is a multilinearity test, which is a procedure of sampling a multivariate function on a small fract... |

135 | Approximating maximum independent sets by excluding subgraphs
- Boppana, Halldorsson
- 1992
(Show Context)
Citation Context ...his work was done while the author was visiting Princeton University. ¶ AT&T Bell Labs, Murray Hill, NJ 07974. 1sThe best upper bound known, is that ω(G) can be approximated within a factor n of log2 =-=[10]-=- in polynomial time. It is natural to ask by how much this upper bound n can be improved, and whether there is some factor within which approximating ω(G) is hard. Proving that problem L is NP-complet... |

129 | On the power of multi-prover interactive protocols
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(Show Context)
Citation Context ...e give a new characterization of NP in the domain of interactive proofs. We show that any language L ∈ NP is accepted by a multi-prover protocol (using the probabilistic oracle machine formulation of =-=[19]-=-) in which the number of random bits used by the verifier and answer bits sent by the oracle is small — O(log n · log log n). To do so, we consider the theorem of Babai, Fortnow and Lund ([5]), showin... |

127 | The monotone circuit complexity of boolean functions
- Alon, Boppana
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(Show Context)
Citation Context ...y, if clique has n ɛ approximation for arbitrarily small ɛ, then all problems in this class have polynomial time constant approximation schemes within factors arbitrarily close to 1. Alon and Boppana =-=[1]-=- show that there is no polynomial time monotone circuit that approximates the size n of the maximum clique in an n-node graph up to a factor of logO(1) n . 1.4 Subsequent Work on Approximation When ou... |

117 |
Does co-NP have short interactive proofs
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(Show Context)
Citation Context ...works are similar. Our work shows an interesting relation between work on interactive proof systems and long standing open problems in computational complexity theory. For earlier examples, see [21], =-=[11]-=-, [18], [14], [13]. 3s1.2 The Multi Linearity Test An important ingrediant in the [5] protocol is a multilinearity test, which is a procedure of sampling a multivariate function on a small fraction of... |

90 | On the power of two points based sampling
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Citation Context ...only O(m log |F |) random bits, generates O(m) sample triples, and ”hits” a non-f-linear triple with probability at least 1/2. Problems of this type can be handled by the method of two-point sampling =-=[12]-=-. The basic idea behind two point sampling techniques is that pairwise independent sample points (or sample triples, in our case) share many of the properties of mutually independent sample points esp... |

73 | A still better performance guarantee for approximate graph coloring
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(Show Context)
Citation Context ...required to color the nodes of a graph so there is no monochromatic edge). It was known that approximating the chromatic number n(log log n)2 log 3 n within an factor is in polynomial time ([34], [8],=-=[24]-=-), while approximating the chromatic number within any factor smaller than 2 is NP-hard [22]. Approxin mating the chromatic number within any factor between 2 and log3 was not known n to be in P or to... |

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On the complexity of approximating the independent set problem
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Citation Context ...results imply that if ˜P approximation algorithms exist for any of the approximation classes that [31] define, or for any of the RMAX(2)-hard approximation problems, then NP ⊂ ˜P. Berman and Schnitger=-=[9]-=- show that approximating ω(G) within factor n ɛ is hard for MAX SNP under randomized reductions. Namely, if clique has n ɛ approximation for arbitrarily small ɛ, then all problems in this class have p... |

68 |
Improving the performance guarantee for approximate graph coloring
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(Show Context)
Citation Context ...of colors required to color the nodes of a graph so there is no monochromatic edge). It was known that approximating the chromatic number n(log log n)2 log 3 n within an factor is in polynomial time (=-=[34]-=-, [8],[24]), while approximating the chromatic number within any factor smaller than 2 is NP-hard [22]. Approxin mating the chromatic number within any factor between 2 and log3 was not known n to be ... |

58 |
Ramsey numbers and an approximation algorithm for the vertex cover problem
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(Show Context)
Citation Context ...x cover problem (the minimum subset of vertices that contains at least one of any two adjacent vertices). log log n The minimum size vertex cover can be approximated within a factor of 2 − Ω( log n ) =-=[6, 30]-=- in polynomial time. No NP-hardness results was known for approximating vertex cover. Papadimitriou and Yannakakis [32] initiated a classification of NP optimization problems based on their logical ch... |

57 |
The complexity of near optimal graph coloring
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(Show Context)
Citation Context ... approximating the chromatic number n(log log n)2 log 3 n within an factor is in polynomial time ([34], [8],[24]), while approximating the chromatic number within any factor smaller than 2 is NP-hard =-=[22]-=-. Approxin mating the chromatic number within any factor between 2 and log3 was not known n to be in P or to be NP-complete. Another example is the vertex cover problem (the minimum subset of vertices... |

54 | On the complexity of space bounded interactive proofs (extended abstract
- Condon, Lipton
- 1989
(Show Context)
Citation Context ...milar. Our work shows an interesting relation between work on interactive proof systems and long standing open problems in computational complexity theory. For earlier examples, see [21], [11], [18], =-=[14]-=-, [13]. 3s1.2 The Multi Linearity Test An important ingrediant in the [5] protocol is a multilinearity test, which is a procedure of sampling a multivariate function on a small fraction of its domain,... |

45 |
Quantifiers and approximation
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(Show Context)
Citation Context ...proximation. Some examples of complete problems in this class are independent set in bounded degree graphs and satisfying the maximum number of clauses in a Boolean(CNF) formula. Panconesi and Ranjan =-=[31]-=- extend [32]’s approach and define the class MAX Π1. The complete problems for MAX Π1 cannot be approximated unless P = NP. Consequently, [31] define subclasses of MAX Π1 for which the approximability... |

25 |
A better performance guarantee for approximate graph coloring
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(Show Context)
Citation Context ...ors required to color the nodes of a graph so there is no monochromatic edge). It was known that approximating the chromatic number n(log log n)2 log 3 n within an factor is in polynomial time ([34], =-=[8]-=-,[24]), while approximating the chromatic number within any factor smaller than 2 is NP-hard [22]. Approxin mating the chromatic number within any factor between 2 and log3 was not known n to be in P ... |

22 |
Multi prover interactive proofs: How to remove intractability assumptions
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- 1988
(Show Context)
Citation Context ...ystem and scale it down to complexity classes lower than NEXPTIME. 2 Multi-Prover Protocols The model of multi-prover interactive proofs was introduced by Ben-Or, Goldwasser, Kilian, and Wigderson in =-=[7]-=-. It is defined as follows. 6sLet P1, P2 be infinitely powerful machines and V be a probabilistic polynomialtime Turing machine, all of which share the same read-only input tape. The verifier V shares... |

21 |
Low-degree tests
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(Show Context)
Citation Context ...on of this paper, we proposed a test with a somewhat different structure, with the same complexity O(m log |F|). Generalizations of this other test and simplification of its analysis are presented in =-=[20]-=-. 17s4.3.1 Developing The Mathematical Background First we introduce some notation; the distance between two functions f1, f2 : F m → F, ∆(f1, f2) = |{�y|f1(�y) �= f2(�y)}| |{all �y}| i.e., the fracti... |

17 |
A.: Multi-oracle interactive protocols with space bounded verifiers
- Feige, Shamir
- 1989
(Show Context)
Citation Context ...are similar. Our work shows an interesting relation between work on interactive proof systems and long standing open problems in computational complexity theory. For earlier examples, see [21], [11], =-=[18]-=-, [14], [13]. 3s1.2 The Multi Linearity Test An important ingrediant in the [5] protocol is a multilinearity test, which is a procedure of sampling a multivariate function on a small fraction of its d... |

15 | Graph products and chromatic numbers - Linial, Vazirani - 1989 |

6 |
On the Complexity of Approximating the Maximum Size of a Clique. Unpublished manuscript
- Feige, Goldwasser, et al.
- 1990
(Show Context)
Citation Context ...who uses O((log n) c ) randomness and answer size (which is sufficient to obtain the result that ω(G) is hard to approximate within a factor of 2 log1−ɛ n for any ɛ > 0, unless NP ⊆ ˜P), was obtained =-=[16]-=- independently from [4] who also scaled down [5]’s protocol to the NP level. In their work, they bound the total running time of the verifier by poly(logarithmic) time rather than the particular param... |

5 |
The complexity of the max word problem
- Condon
- 1991
(Show Context)
Citation Context ... Our work shows an interesting relation between work on interactive proof systems and long standing open problems in computational complexity theory. For earlier examples, see [21], [11], [18], [14], =-=[13]-=-. 3s1.2 The Multi Linearity Test An important ingrediant in the [5] protocol is a multilinearity test, which is a procedure of sampling a multivariate function on a small fraction of its domain, and u... |

2 | A 2 \Gamma log log n 2 log n performance ratio for the weighted vertex cover problem - Bar-Yehuda, Even - 1983 |

1 | A 2 − performance ratio for the weighted 2 log n vertex cover problem - Bar-Yehuda, Even - 1983 |