The complexity of somewhat approximation resistant predicates

Subhash Khot, Madhur Tulsiani, Pratik Worah

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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)|/ 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.

Original languageEnglish (US)
Title of host publicationAutomata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings
PublisherSpringer Verlag
Pages689-700
Number of pages12
Volume8572 LNCS
EditionPART 1
ISBN (Print)9783662439470
DOIs
StatePublished - 2014
Event41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark
Duration: Jul 8 2014Jul 11 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume8572 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other41st International Colloquium on Automata, Languages, and Programming, ICALP 2014
CountryDenmark
CityCopenhagen
Period7/8/147/11/14

Fingerprint

Predicate
Hardness
Hamming distance
Hamming Distance
Assignment
Approximation
Integrality
G-function
Approximation algorithms
Beat
Supremum
Approximation Algorithms
Strictly
NP-complete problem
Denote
Output
Coefficient
Hierarchy

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Khot, S., Tulsiani, M., & Worah, P. (2014). The complexity of somewhat approximation resistant predicates. In 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.

Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings. Vol. 8572 LNCS PART 1. ed. Springer Verlag, 2014. p. 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).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Khot, S, Tulsiani, M & Worah, P 2014, The complexity of somewhat approximation resistant predicates. in 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
Khot S, Tulsiani M, Worah P. The complexity of somewhat approximation resistant predicates. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings. PART 1 ed. Vol. 8572 LNCS. Springer Verlag. 2014. p. 689-700. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1). https://doi.org/10.1007/978-3-662-43948-7_57
Khot, Subhash ; Tulsiani, Madhur ; Worah, Pratik. / The complexity of somewhat approximation resistant predicates. Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings. Vol. 8572 LNCS PART 1. ed. Springer Verlag, 2014. pp. 689-700 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1).
@inproceedings{8f1020871b884077bd4cf09342a643ef,
title = "The complexity of somewhat approximation resistant predicates",
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)|/ 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.",
author = "Subhash Khot and Madhur Tulsiani and Pratik Worah",
year = "2014",
doi = "10.1007/978-3-662-43948-7_57",
language = "English (US)",
isbn = "9783662439470",
volume = "8572 LNCS",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
number = "PART 1",
pages = "689--700",
booktitle = "Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings",
edition = "PART 1",

}

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 -