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The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs
unpublished
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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]
2018
arXiv
pre-print
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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
2019
Foundations and Trends® in Theoretical Computer Science
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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
2016
ACM Transactions on Computational Logic
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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
2019
Computational Complexity Conference
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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]
2014
arXiv
pre-print
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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
2014
2014 IEEE 29th Conference on Computational Complexity (CCC)
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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]
2018
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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)
2018
Dagstuhl Reports
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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]
2013
Lecture Notes in Computer Science
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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]
2022
arXiv
pre-print
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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
2018
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18
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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)
2018
Dagstuhl Reports
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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]
2019
arXiv
pre-print
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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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