The SDP Value for Random Two-Eigenvalue CSPs

Authors Sidhanth Mohanty, Ryan O'Donnell, Pedro Paredes

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Sidhanth Mohanty
  • EECS Department, University of California, Berkeley, CA, USA
Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Pedro Paredes
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA


We thank Yuval Peled for emphasizing the bipartite graph view of additive lifts, and Tselil Schramm for helpful discussions surrounding the trace method on graphs. S.M. would like to thank Jess Banks and Prasad Raghavendra for plenty of helpful discussions on orthogonal polynomials and nonbacktracking walks. Finally, we are grateful to Xinyu Wu for bringing the relevance of [Bordenave and Collins, 2018] to our attention and helping us to understand the issues discussed in Section A.

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Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. The SDP Value for Random Two-Eigenvalue CSPs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 50:1-50:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, "two-eigenvalue 2CSPs". We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular 2XOR and NAE-3SAT, and includes new cases such as random Sort₄ (equivalently, CHSH) and Forrelation CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara-Bass Formula, and the Friedman/Bordenave proof of Alon’s Conjecture.

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ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Semidefinite programming
  • constraint satisfaction problems


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