Abstract
It has often been claimed in recent papers that one can find a degree d SumofSquares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O'Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits lowdegree proofs of nonnegativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs.
First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common usecases of the SoS algorithm, such as MaxCSP, Balanced Separator, MaxClique, MaxBisection, and UnitVector constraints.
On the negative side, O'Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and nonnegative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n).
BibTeX  Entry
@InProceedings{raghavendra_et_al:LIPIcs:2017:7380,
author = {Prasad Raghavendra and Benjamin Weitz},
title = {{On the Bit Complexity of SumofSquares Proofs}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {80:180:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7380},
URN = {urn:nbn:de:0030drops73804},
doi = {10.4230/LIPIcs.ICALP.2017.80},
annote = {Keywords: SumofSquares, Combinatorial Optimization, Proof Complexity}
}
Keywords: 

SumofSquares, Combinatorial Optimization, Proof Complexity 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 