Explicit SoS Lower Bounds from High-Dimensional Expanders

Authors Irit Dinur, Yuval Filmus , Prahladh Harsha , Madhur Tulsiani

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Author Details

Irit Dinur
  • Weizmann Institute of Science, Rehovot, Israel
Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Madhur Tulsiani
  • Toyota Technological Institute at Chicago, IL, USA


Part of this work was done when the authors were visiting the Simons Institute of Theory of Computing, Berkeley for the 2019 summer cluster on "Error-Correcting Codes and High-Dimensional Expansion". We thank the Simons Institute for their kind hospitality.

Cite AsGet BibTex

Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit SoS Lower Bounds from High-Dimensional Expanders. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We construct an explicit and structured family of 3XOR instances which is hard for O(√{log n}) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems are highly structured and can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author (FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Proof complexity
  • High-dimensional expanders
  • sum-of-squares
  • integrality gaps


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