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SOS Lower Bounds with Hard Constraints: Think Global, Act Local

Authors Pravesh K. Kothari, Ryan O'Donnell, Tselil Schramm

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ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • sum-of-squares hierarchy
  • random constraint satisfaction problems

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Abstract

Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value beta, it did not necessarily actually "satisfy" the constraint "objective = beta". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating global constraints into local constraints. Using these ideas, we also show that degree-Omega(sqrt{n}) SOS does not provide a (4/3 - epsilon)-approximation for Min-Bisection, and degree-Omega(n) SOS does not provide a (11/12 + epsilon)-approximation for Max-Bisection or a (5/4 - epsilon)-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.

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Pravesh K. Kothari, Ryan O'Donnell, and Tselil Schramm. SOS Lower Bounds with Hard Constraints: Think Global, Act Local. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.49

Author Details

Pravesh K. Kothari
  • Department of Computer Science, Princeton University and Institute for Advanced Study, Princeton, USA
Ryan O'Donnell
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA
Tselil Schramm
  • Department of Computer Science, Harvard and MIT, Cambridge, USA

Funding

  • O'Donnell, Ryan: Some work performed at the Boğaziçi University Computer Engineering Department, supported by Marie Curie International Incoming Fellowship project number 626373. Also supported by NSF grants CCF-1618679, CCF-1717606. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).
  • Schramm, Tselil: This work was partly supported by an NSF Graduate Research Fellowship (1106400), and also by a Simons Institute Fellowship.

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