Strongly refuting all semi-random Boolean CSPs

Jackson Abascal, Venkatesan Guruswami, Pravesh K. Kothari

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

4 Scopus citations


We give an efficient algorithm to strongly refute semirandom instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean k-XOR problem on n variables that have Oe(nk/2) constraints. (In a semi-random k-XOR instance, the equations can be arbitrary and only the right hand sides are random.) One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work. Our approach involves taking an instance that does not satisfy this property (i.e., is not pseudorandom) and reducing it to a partitioned collection of 2-XOR instances. We analyze these subinstances using a carefully chosen quadratic form as proxy, which in turn is bounded via a combination of spectral methods and semidefinite programming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernstein inequality. Even for the purely random case, this leads to a shorter proof compared to the ones in the literature that rely on problem-specific trace-moment computations.

Original languageEnglish (US)
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Number of pages19
ISBN (Electronic)9781611976465
StatePublished - 2021
Externally publishedYes
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: Jan 10 2021Jan 13 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


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