Tight size-degree bounds for sums-of-squares proofs
Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
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 [Krajícek'04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, Müller, and Oliva'13] and [Atserias, Lauria, and Nordström'14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.
Original language | English |
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Title of host publication | 30th Conference on Computational Complexity, CCC 2015 |
Editors | David Zuckerman |
Number of pages | 19 |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Publication date | 1 Jun 2015 |
Pages | 448-466 |
ISBN (Electronic) | 9783939897811 |
DOIs | |
Publication status | Published - 1 Jun 2015 |
Externally published | Yes |
Event | 30th Conference on Computational Complexity, CCC 2015 - Portland, United States Duration: 17 Jun 2015 → 19 Jun 2015 |
Conference
Conference | 30th Conference on Computational Complexity, CCC 2015 |
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Land | United States |
By | Portland |
Periode | 17/06/2015 → 19/06/2015 |
Sponsor | Microsoft Research |
Series | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 33 |
ISSN | 1868-8969 |
- Clique, Degree, Lasserre, Lower bound, Positivstellensatz, Proof complexity, Rank, Resolution, Semidefinite programming, Size, SOS, Sums-ofsquares
Research areas
ID: 251869202