### Abstract

A boolean predicate f:{0,1}^{k} → {0,1} is said to be somewhat approximation resistant if for some constant τ > |f^{-1}(1)|/ 2^{k}, given a τ-satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum over all τ for which this holds. It is known that a predicate is somewhat approximation resistant precisely when its Fourier degree is at least 3. For such predicates, we give a characterization of the hardness gap (τ(f)-|f^{-1}(1)|/2^{k}) up to a factor of O(k^{5}). We show that the hardness gap is determined by two factors: - The nearest Hamming distance of f to a function g of Fourier degree at most 2, which is related to the Fourier mass of f on coefficients of degree 3 or higher. - Whether f is monotonically below g. When the Hamming distance is small and f is monotonically below g, we give an SDP-based approximation algorithm and hardness results otherwise. We also give a similar characterization of the integrality gap for the natural SDP relaxation of MAX k-CSP(f) after Ω(n) rounds of the Lasserre hierarchy.

Original language | English (US) |
---|---|

Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 689-700 |

Number of pages | 12 |

Volume | 8572 LNCS |

Edition | PART 1 |

ISBN (Print) | 9783662439470 |

DOIs | |

State | Published - 2014 |

Event | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark Duration: Jul 8 2014 → Jul 11 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Number | PART 1 |

Volume | 8572 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 |
---|---|

Country | Denmark |

City | Copenhagen |

Period | 7/8/14 → 7/11/14 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings*(PART 1 ed., Vol. 8572 LNCS, pp. 689-700). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8572 LNCS, No. PART 1). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_57

**The complexity of somewhat approximation resistant predicates.** / Khot, Subhash; Tulsiani, Madhur; Worah, Pratik.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings.*PART 1 edn, vol. 8572 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 8572 LNCS, Springer Verlag, pp. 689-700, 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014, Copenhagen, Denmark, 7/8/14. https://doi.org/10.1007/978-3-662-43948-7_57

}

TY - GEN

T1 - The complexity of somewhat approximation resistant predicates

AU - Khot, Subhash

AU - Tulsiani, Madhur

AU - Worah, Pratik

PY - 2014

Y1 - 2014

N2 - A boolean predicate f:{0,1}k → {0,1} is said to be somewhat approximation resistant if for some constant τ > |f-1(1)|/ 2k, given a τ-satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum over all τ for which this holds. It is known that a predicate is somewhat approximation resistant precisely when its Fourier degree is at least 3. For such predicates, we give a characterization of the hardness gap (τ(f)-|f-1(1)|/2k) up to a factor of O(k5). We show that the hardness gap is determined by two factors: - The nearest Hamming distance of f to a function g of Fourier degree at most 2, which is related to the Fourier mass of f on coefficients of degree 3 or higher. - Whether f is monotonically below g. When the Hamming distance is small and f is monotonically below g, we give an SDP-based approximation algorithm and hardness results otherwise. We also give a similar characterization of the integrality gap for the natural SDP relaxation of MAX k-CSP(f) after Ω(n) rounds of the Lasserre hierarchy.

AB - A boolean predicate f:{0,1}k → {0,1} is said to be somewhat approximation resistant if for some constant τ > |f-1(1)|/ 2k, given a τ-satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum over all τ for which this holds. It is known that a predicate is somewhat approximation resistant precisely when its Fourier degree is at least 3. For such predicates, we give a characterization of the hardness gap (τ(f)-|f-1(1)|/2k) up to a factor of O(k5). We show that the hardness gap is determined by two factors: - The nearest Hamming distance of f to a function g of Fourier degree at most 2, which is related to the Fourier mass of f on coefficients of degree 3 or higher. - Whether f is monotonically below g. When the Hamming distance is small and f is monotonically below g, we give an SDP-based approximation algorithm and hardness results otherwise. We also give a similar characterization of the integrality gap for the natural SDP relaxation of MAX k-CSP(f) after Ω(n) rounds of the Lasserre hierarchy.

UR - http://www.scopus.com/inward/record.url?scp=84904216933&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904216933&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_57

DO - 10.1007/978-3-662-43948-7_57

M3 - Conference contribution

AN - SCOPUS:84904216933

SN - 9783662439470

VL - 8572 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 689

EP - 700

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

ER -