Tight Size-Degree Bounds for Sums-of-Squares Proofs
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Tight Size-Degree Bounds for Sums-of-Squares Proofs. / Lauria, Massimo; Nordström, Jakob.
In: Computational Complexity, Vol. 26, No. 4, 01.12.2017, p. 911-948.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Tight Size-Degree Bounds for Sums-of-Squares Proofs
AU - Lauria, Massimo
AU - Nordström, Jakob
PY - 2017/12/1
Y1 - 2017/12/1
N2 - We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size nΩ ( d ) for values of d = d(n) from constant all the way up to nδ for some universal constant δ. This shows that the nO ( d ) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajíček (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.
AB - We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size nΩ ( d ) for values of d = d(n) from constant all the way up to nδ for some universal constant δ. This shows that the nO ( d ) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajíček (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.
KW - clique
KW - degree
KW - Lasserre
KW - lower bound
KW - Positivstellensatz
KW - Proof complexity
KW - rank
KW - resolution
KW - semidefinite programming
KW - size
KW - SOS
KW - sums-of-squares
UR - http://www.scopus.com/inward/record.url?scp=85018513275&partnerID=8YFLogxK
U2 - 10.1007/s00037-017-0152-4
DO - 10.1007/s00037-017-0152-4
M3 - Journal article
AN - SCOPUS:85018513275
VL - 26
SP - 911
EP - 948
JO - Computational Complexity
JF - Computational Complexity
SN - 1016-3328
IS - 4
ER -
ID: 251868223