## Gowers uniformity, influence of variables, and PCPs (2006)

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Venue: | In Proceedings of the 38th Annual ACM Symposium on Theory of Computing |

Citations: | 51 - 2 self |

### BibTeX

@INPROCEEDINGS{Samorodnitsky06gowersuniformity,,

author = {Alex Samorodnitsky and Luca Trevisan},

title = {Gowers uniformity, influence of variables, and PCPs},

booktitle = {In Proceedings of the 38th Annual ACM Symposium on Theory of Computing},

year = {2006},

pages = {11--20}

}

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### Abstract

Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimension-d uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimension-d Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that |〈{fS} 〉 U d | ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph long-code test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 1

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Citation Context ... their applications to the study of the approximability of optimization problems are too many to summarize here, and we refer the reader to the chapter on hardness of approximation in Vazirani’s book =-=[Vaz01]-=- and to some recent survey papers [Aro02, Fei02, Tre04]. In this paper we are interested in the following question: for a given number of queries, what is the highest confidence that we can have in th... |

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Citation Context ... by Engebretsen and Holmerin [EH05]. As we discuss below, 2Θ( √ q) /2q was a natural limit for the soundness achievable with current techniques. In this paper, assuming Khot’s Unique Games Conjecture =-=[Kho02]-=-, we present an improvement to s = (q + 1)/2 q . Our analysis is based on a theorem, which is probably of independent interest, bounding the Gowers uniformity of a given function in terms of the influ... |

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Citation Context ... length-k progression. This, unfortunately, does not seem to be true, and sets whose characteristic function is defined in terms of a degree2 polynomial are basic counterexamples even for k = 4. (See =-=[Gow01]-=-.) Recall that something similar happens in the hypergraph test, where a function defined in terms of a degree-2 polynomial is very far from linear (and so all its Fourier coefficients are small), but... |

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Citation Context ...ing a lower bound of 1 + (1 − o(1)) log q q to the amortized query complexity of a q-query PCP.) The PCP Theorem shows that we can have s = 1/2Ω(q) , and the authors showed that can have s ≤ 22√q /2q =-=[ST00]-=-. (That is, the amortized query complexity can be as low as 1+O(1/ √ q).) Our proof was simplified by H˚astad and Wigderson [HW03], and the soundness was improved to s ≤ 2 √ 2q /2q by Engebretsen and ... |

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Citation Context ...em 29 that, assuming the unique games conjecture, Max kCSP cannot be approximated within a factor larger than (k + 1)/2 k if k is of the form 2 t − 1. It follows from Theorem 29 and the reductions in =-=[Tre01]-=- that, assuming the unique games conjecture, the Maximum Independent Set problem in graphs of maximum degree D cannot be approximated within a factor larger (log D) c /D, for sufficiently large D, whe... |

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Citation Context ...s, the amortized query complexity can be as low as 1+O(1/ √ q).) Our proof was simplified by H˚astad and Wigderson [HW03], and the soundness was improved to s ≤ 2 √ 2q /2q by Engebretsen and Holmerin =-=[EH05]-=-. As we discuss below, 2Θ( √ q) /2q was a natural limit for the soundness achievable with current techniques. In this paper, assuming Khot’s Unique Games Conjecture [Kho02], we present an improvement ... |

19 |
An inverse theorem for the Gowers U 3 norm
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Citation Context ...ctural properties of functions f : ZN → C, N prime, having non-trivially large U d value. Such functions are shown to have a certain “local correlation” with degree (d − 1)-polynomials. Green and Tao =-=[GT05]-=- study functions f : G → C with large U 3 , and, provided that the order of G is not divided by and 2 and 3, prove a certain “global correlation” between such functions and degree-2 polynomials. 3 Sam... |

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and Avi Wigderson. Simple analysis of graph tests for linearity and PCP. Random Struct
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Citation Context ... have s = 1/2Ω(q) , and the authors showed that can have s ≤ 22√q /2q [ST00]. (That is, the amortized query complexity can be as low as 1+O(1/ √ q).) Our proof was simplified by H˚astad and Wigderson =-=[HW03]-=-, and the soundness was improved to s ≤ 2 √ 2q /2q by Engebretsen and Holmerin [EH05]. As we discuss below, 2Θ( √ q) /2q was a natural limit for the soundness achievable with current techniques. In th... |

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Citation Context ..., that is, the amortized query complexity must be at least 1. A more careful argument gives a lower bound of 2/2q [Tre98a] on the soundness, which was recently improved to Ω( q 1 log q · 2q ) by Hast =-=[Has05]-=-. (Hast’s result can also be stated as giving a lower bound of 1 + (1 − o(1)) log q q to the amortized query complexity of a q-query PCP.) The PCP Theorem shows that we can have s = 1/2Ω(q) , and the ... |

7 | Linear consistency testing
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Citation Context ...unction There is a natural extension of the BLR test to this setting: 3-functions-BLR-Test (f,g,h) choose x,y uniformly at random in {0,1} n accept if and only if f(x) · g(y) = h(x + y) Aumann et al. =-=[AHRS01]-=- show that if this test accepts with probability 1/2 + ε, then there is a linear function χS such that f,g,h have all agreement at least 1/2 + ε/3 with either χS or −χS. The second change is that, for... |

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Hypergraph linearity and quadraticity tests for boolean functions, preprint
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Citation Context ...tions f : G → C with large U 3 , and, provided that the order of G is not divided by and 2 and 3, prove a certain “global correlation” between such functions and degree-2 polynomials. 3 Samorodnitsky =-=[Sam05]-=- proves such a result for functions f : {0,1} n → R. Not much is known about functions f : G → C having large U d when d ≥ 4 and G is a general group. 1.3 Our Results We prove that if f : G1 ×· · ·×Gn... |

3 | YUVAL RABANI, AND D.SIVAKUMAR: On the hardness of approximating multicut and sparsest-cut - CHAWLA, KRAUTHGAMER, et al. - 2005 |

1 | NP got a new definition: a survey of probabilistically checkable proofs - How - 2002 |