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Sum-Of-Squares Bounds via Boolean Function Analysis

Author Adam Kurpisz

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Author Details

Adam Kurpisz
  • ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

Funding

Supported by SNSF project PZ00P2_174117.

Acknowledgements

I would like to express my gratitude to Markus Schweighofer for fruitful discussions.

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Adam Kurpisz. Sum-Of-Squares Bounds via Boolean Function Analysis. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.79

Abstract

We introduce a method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory. We apply our technique to improve upon existing results, thus making progress towards answering several open questions.
We consider two questions by Laurent. First, finding what is the SoS rank of the linear representation of the set with no integral points. We prove that the SoS rank is between ceil[n/2] and ceil[~ n/2 +sqrt{n log{2n}} ~]. Second, proving the bounds on the SoS rank for the instance of the Min Knapsack problem. We show that the SoS rank is at least Omega(sqrt{n}) and at most ceil[{n+ 4 ceil[sqrt{n} ~]}/2]. Finally, we consider the question by Bienstock regarding the instance of the Set Cover problem. For this problem we prove the SoS rank lower bound of Omega(sqrt{n}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Convex optimization
Keywords
  • SoS certificate
  • SoS rank
  • hypercube optimization
  • semidefinite programming

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