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Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems

Stefan S. Dantchev
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
We prove a dichotomy theorem for the rank of the uniformly generated (i.e. expressible in First-Order (FO) Logic) propositional tautologies in both the Lovász-Schrijver (LS) and Sherali-Adams (SA) proof  ...  Up to now, this kind of so-called "Complexity Gap" theorems have been known for Tree-like Resolution and, in somehow restricted forms, for the Resolution and Nullstellensatz proof systems.  ...  The Sherali-Adams (SA) proof system is a static proof system, so we shall define SA proofs of rank k for every k, 0 ≤ k < n.  ... 
doi:10.1145/1250790.1250837 dblp:conf/stoc/Dantchev07 fatcat:dmce3ncwcndv3aljwtkzvsfeyi

Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems

Stefan Dantchev, Barnaby Martin
2012 Computational Complexity  
We prove a dichotomy theorem for the rank of propositional contradictions, uniformly generated from first-order sentences, in both the Lovász-Schrijver (LS) and Sherali-Adams (SA) refutation systems.  ...  Until now, this kind of so-called "complexity gap" theorem has been known for tree-like Resolution and, in somehow restricted forms, for the Resolution and Nullstellensatz systems.  ...  We are grateful to two anonymous referees for interesting observations as well as detailed corrections.  ... 
doi:10.1007/s00037-012-0049-1 fatcat:4judqku2tnem7cprdklf3z6mme

Sherali-adams relaxations of the matching polytope

Claire Mathieu, Alistair Sinclair
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
We study the Sherali-Adams lift-and-project hierarchy of linear programming relaxations of the matching polytope.  ...  We also show that the rank of the matching polytope (i.e., the number of Sherali-Adams rounds until the integer polytope is reached) is exactly 2d − 1.  ...  Theorem 1.3 answers for the Sherali-Adams hierarchy a question initially posed by Lovász and Schrijver about the rank of the matching polytope in the LS+ hierarchy, which was answered by Stephen and Tunçel  ... 
doi:10.1145/1536414.1536456 dblp:conf/stoc/MathieuS09 fatcat:4qkzz7y6tbghpjxtobd42jxss4

Proof Complexity Meets Algebra

Albert Atserias, Joanna Ochremiak
2018 ACM Transactions on Computational Logic  
We give an example of such a proof system by showing that bounded-degree Lovász-Schrijver satisfies both requirements.  ...  We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a core preserve  ...  The semi-algebraic proof systems for which we prove it include Sherali-Adams, Lasserre/SOS, and Lovász-Schrijver of bounded and unbounded degree.  ... 
doi:10.1145/3265985 fatcat:hhamzk75vfhrbj4vuf7eo5lzmm

On the Rank of Cutting-Plane Proof Systems [chapter]

Sebastian Pokutta, Andreas S. Schulz
2010 Lecture Notes in Computer Science  
In addition, the rank of the Sherali-Adams cuts (for fixed level d), the rank of the linear matrix cuts of Lovász and Schrijver, the rank of the lift-and-project cuts by Balas, Ceria, and Cornuéjols, and  ...  Lower bounds of n for the matrix cut operators N 0 , N, and N + of Lovász and Schrijver Lovász and Schrijver [1991] were given in Cook and Dash [2001] , Cornuéjols and Li [2002b] , Goemans and Tuncel [  ...  We now consider the cutting-plane procedures N 0 and N, introduced by Lovász and Schrijver Lovász and Schrijver [1991] (see also Balas et al. [1993] for N 0 and Sherali and Adams [1990] for N), which  ... 
doi:10.1007/978-3-642-13036-6_34 fatcat:626wcllt6fbczbjdj5zv557zo4

Proof Complexity Meets Algebra [article]

Albert Atserias, Joanna Ochremiak
2018 arXiv   pre-print
We give an example of such a proof system by showing that bounded-degree Lov\'asz-Schrijver satisfies both requirements.  ...  We show that, for the most studied propositional, algebraic, and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a  ...  We are grateful to Massimo Lauria for his help in reconstructing the proofs in Section 2.4, and for several other insightful comments during the development of this work.  ... 
arXiv:1711.07320v2 fatcat:r43hyaabzrbvvj42qiyplxguyy

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordstrom
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well.  ...  Such variables were missing in the original definition in [CEI96] but adding them makes for a more natural and well-behaved proof system.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
doi:10.1109/ccc.2014.36 dblp:conf/coco/AtseriasLN14 fatcat:uek3kmbdobaehdcplxysvsmoaa

Narrow Proofs May Be Maximally Long [article]

Albert Atserias, Massimo Lauria, Jakob Nordström
2014 arXiv   pre-print
Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well.  ...  Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
arXiv:1409.2731v1 fatcat:luvihb4my5dnnmegzob5lomsne

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordström
2016 ACM Transactions on Computational Logic  
We refer to [Lau01, CT12] for a more detailed discussion of Sherali-Adams,  ...  Such a certificate is a rank-k Sherali-Adams refutation of the corresponding CNF formula.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work.  ... 
doi:10.1145/2898435 fatcat:fyixt7jbinastnhcaltahbrjmi

Rank bounds for a hierarchy of Lovász and Schrijver

Pratik Worah
2013 Journal of combinatorial optimization  
In this paper we show that LS * rank is incomparable to other hierarchies like LS + and Sherali-Adams (SA) and show rank lower bounds for P HP n+1 n and integrality gaps for optimization problems like  ...  Lovász and Schrijver [17] introduced several lift and project methods for 0-1 integer programs, now collectively known as Lovász-Schrijver (LS) hierarchies.  ...  Acknowledgements The author thanks Yury Makarychev for reading several drafts of this paper and also for his help with proofs in Section 6.  ... 
doi:10.1007/s10878-013-9662-4 fatcat:7xzx3bo4wbaorlej3t4ezbyfqy

DRL*: A hierarchy of strong block-decomposable linear relaxations for 0–1 MIPs

M. Minoux, H. Ouzia
2010 Discrete Applied Mathematics  
(L&P) hierarchy and the Sherali-Adams (RLT) hierarchy.  ...  As an application, we show that both the RLT and DRL* relaxations are the same up to rank d for the problem of optimizing a pseudoboolean function of degree d over a polyhedron.  ...  Acknowledgements We gratefully acknowledge three referees for their many constructive comments and suggestions.  ... 
doi:10.1016/j.dam.2010.08.020 fatcat:brwhj6ralrgjnitm5bomcyi2jm

Approximate Constraint Satisfaction Requires Large LP Relaxations [article]

Siu On Chan and James R. Lee and Prasad Raghavendra and David Steurer
2016 arXiv   pre-print
In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8.  ...  We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy.  ...  Acknowledgements We thank the anonymous referees for many useful suggestions and observations. S. O. Chan was supported by NSF grants CCF-1118083 and CCF-1017403. P.  ... 
arXiv:1309.0563v3 fatcat:a6amyvzzhzfwtoypxttewgjkee

Approximation Limits of Linear Programs (Beyond Hierarchies)

Gabor Braun, Samuel Fiorini, Sebastian Pokutta, David Steurer
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science  
We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices.  ...  Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts  ...  Adams [1990] and Lovász and Schrijver [1991] ).  ... 
doi:10.1109/focs.2012.10 dblp:conf/focs/BraunFPS12 fatcat:ue7kjcuvirctpbs667cutcricm

Approximation Limits of Linear Programs (Beyond Hierarchies)

Gábor Braun, Samuel Fiorini, Sebastian Pokutta, David Steurer
2015 Mathematics of Operations Research  
We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices.  ...  Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts  ...  Adams [1990] and Lovász and Schrijver [1991] ).  ... 
doi:10.1287/moor.2014.0694 fatcat:gimzdyx7nnfdlp5gqkbpeycfmy

Hardness amplification in proof complexity

Paul Beame, Trinh Huynh, Toniann Pitassi
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
k)), as well as Sherali-Adams and Lasserre proofs.  ...  As special cases, such systems include: Lovász-Schrijver systems (LS, LS + ), high degree analogues of Lovász-Schrijver (LS(k), LS + (k)), Cutting Planes and high degree versions of Cutting Planes (CP(  ...  Acknowledgements We would like to thank Ran Raz and Rahul Santhanam for very helpful conversations. In particular we thank Rahul for suggesting the term hardness escalation.  ... 
doi:10.1145/1806689.1806703 dblp:conf/stoc/BeameHP10 fatcat:os26y7gw5ndkjlq4sba27he77y
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