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The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs

Christoph Berkholz
unpublished
On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations.  ...  Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial  ...  polynomial calculus Sherali-Adams Figure 1 Relation between the proof systems.  ... 
fatcat:hpbkmautyrf4lhttgyu5tqfm7m

Proof Complexity Meets Algebra [article]

Albert Atserias, Joanna Ochremiak
2018 arXiv   pre-print
This class is also known to coincide with the class of languages with refutations of sublinear degree in Sums-of-Squares and Polynomial Calculus over the real-field, for which we provide alternative proofs  ...  We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity.  ...  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

Semialgebraic Proofs and Efficient Algorithm Design

Noah Fleming, Pravesh Kothari, Toniann Pitassi
2019 Foundations and Trends® in Theoretical Computer Science  
We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, Acknowledgments The authors are grateful to Aleksandar Nikolov, Robert Robere, and Morgan  ...  The authors would like to thank Ian Mertz for his suggestions on the presentation of the 3XOR lower bound in Section 5.1.  ...  Most notably are the proof systems Polynomial Calculus (PC) which is gives rise to a family of algebraic algorithms, Sherali-Adams (SA) which gives rise to a large family of linear programs, and Sum-of-Squares  ... 
doi:10.1561/0400000086 fatcat:sotovlogzjfy7kxxloglpu5tym

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordström
2016 ACM Transactions on Computational Logic  
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.  ...  Strong trade-offs between size and space as well as between degree and space have been shown in [BNT13], but-again in analogy with resolution-the exact relations between size and degree remains unclear  ...  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

Sherali - Adams Strikes Back

Ryan O'Donnell, Tselil Schramm, Michael Wagner
2019 Computational Complexity Conference  
We show that the exp c log n log ∆ -round Sherali-Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1% (in fact, at most 1 2 + 2 −Ω(c) ).  ...  Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for max-cut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation  ...  We also thank the Schloss Dagstuhl Leibniz Center for Informatics (and more specifically the organizers of the CSP Complexity and Approximability workshop), as well as the Casa Mathemática  ... 
doi:10.4230/lipics.ccc.2019.8 dblp:conf/coco/ODonnellS19 fatcat:y4qmkupv6ngq3cuvgpmpp2556q

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 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.  ...  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. * This is the full-length version of the paper  ...  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

Sherali--Adams Strikes Back [article]

Ryan O'Donnell, Tselil Schramm
2018 arXiv   pre-print
We show that the (c n/Δ)-round Sherali--Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1% (in fact, at most 12 + 2^-Ω(c)).  ...  Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation.  ...  T.S. also thanks the Oberwolfach Research Institute for Mathematics (and the organizers of the Proof Complexity and Beyond workshop), the Simons Institute (and the organizers of the Optimization semester  ... 
arXiv:1812.09967v1 fatcat:smqmhumghbghvksjejhsje2k7q

Limitations of Algebraic Approaches to Graph Isomorphism Testing [article]

Christoph Berkholz, Martin Grohe
2015 arXiv   pre-print
We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus.  ...  We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations.  ...  Acknowledgements We thank Anuj Dawar and Erkal Selman for many inspiring discussions in the initial phase of this project.  ... 
arXiv:1502.05912v1 fatcat:7ccppsernrhhvfhh77umjkj3yq

The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

Martin Grohe, Venkatesan Guruswami, Stanislav Zivny, Michael Wagner
2018 Dagstuhl Reports  
CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena.  ...  ., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable,  ...  Combining this with our result also implies that any polynomial size LP extended formulation is no stronger than the basic LP, which can be thought of as the base level of the Sherali-Adams hierarchy.  ... 
doi:10.4230/dagrep.8.6.1 dblp:journals/dagstuhl-reports/GroheGZ18 fatcat:3bqo62ly3rgzlnh3bmkvwbuwea

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs [chapter]

Albert Atserias
2013 Lecture Notes in Computer Science  
The inference rules are typically some more or less obvious, non-interesting, and polynomially checkable ways of producing some logical consequence of the hypotheses.  ...  These include resolution or DNF-resolution, which work with clauses and DNFformulas, respectively, and semi-algebraic proofs, which work with polynomial inequalities over the reals.  ...  Acknowledgments We thank the comments of Allen Van Gelder and an anonymous referee on the preliminary draft of this paper.  ... 
doi:10.1007/978-3-642-39071-5_1 fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m

Separations in Proof Complexity and TFNP [article]

Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Tao
2022 arXiv   pre-print
It is well-known that Resolution proofs can be efficiently simulated by Sherali-Adams (SA) proofs.  ...  In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s.  ...  Acknowledgements We thank Albert Atserias, Ilario Bonacina, Pritish Kamath, and David Steurer for discussions.  ... 
arXiv:2205.02168v2 fatcat:lv4bsib3dza3zb6vu46pmva7ny

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

Albert Atserias, Joanna Ochremiak
2018 Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18  
By moving to the duals, our results can be read in terms of Sums-of-Squares (SOS) and Sherali-Adams (SA) proofs, and used to get consequences for Polynomial Calculus (PC) proofs as a side bonus.  ...  As a corollary we get that, for the isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to  ...  We are grateful to Christoph Berkholz, Anuj Dawar, and Wied Pakusa, for useful discussions at an early stage of this work.  ... 
doi:10.1145/3209108.3209186 dblp:conf/lics/AtseriasO18 fatcat:4pxjokuxwzch5oakuwube2l7j4

Proof Complexity (Dagstuhl Seminar 18051)

Albert Atserias, Jakob Nordström, Pavel Pudlák, Rahul Santhanam, Michael Wagner
2018 Dagstuhl Reports  
The study of proof complexity was initiated in [Cook and Reckhow 1979] as a way to attack the P vs.  ...  This workshop, gathering researchers from different strands of the proof complexity community, gave opportunities to take stock of where we stand and discuss the way ahead.  ...  -Proof Complexity Russell Impagliazzo, Pavel Pudlák, and Jiří Sgall. Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Computational Complexity, 8(2):127-144, 1999.  ... 
doi:10.4230/dagrep.8.1.124 dblp:journals/dagstuhl-reports/AtseriasNPS18 fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu

High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts [article]

Monaldo Mastrolilli
2019 arXiv   pre-print
Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/Sum-of-Squares SOS hierarchy fails to capture.  ...  Moreover, for a class of polytopes (e.g. set covering and packing problems), the resulting SOS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up  ...  A polynomial p ∈ R[x] is a sum of squares (SoS) if it can be written as the sum of squares of some other polynomials.  ... 
arXiv:1709.07966v7 fatcat:ns2h3t4qwzaavc6sgdfpcvyuyu
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