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

unpublished

On

fatcat:hpbkmautyrf4lhttgyu5tqfm7m
*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. ...##
###
Proof Complexity Meets Algebra
[article]

2018
*
arXiv
*
pre-print

This class is also known to coincide with

arXiv:1711.07320v2
fatcat:r43hyaabzrbvvj42qiyplxguyy
*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. ...##
###
Semialgebraic Proofs and Efficient Algorithm Design

2019
*
Foundations and Trends® in Theoretical Computer Science
*

We rigorously develop

doi:10.1561/0400000086
fatcat:sotovlogzjfy7kxxloglpu5tym
*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*...##
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Narrow Proofs May Be Maximally Long

2016
*
ACM Transactions on Computational Logic
*

Moreover, our lower bound generalizes to

doi:10.1145/2898435
fatcat:fyixt7jbinastnhcaltahbrjmi
*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. ...##
###
Sherali - Adams Strikes Back

2019
*
Computational Complexity Conference
*

We show that

doi:10.4230/lipics.ccc.2019.8
dblp:conf/coco/ODonnellS19
fatcat:y4qmkupv6ngq3cuvgpmpp2556q
*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 ...##
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Narrow Proofs May Be Maximally Long
[article]

2014
*
arXiv
*
pre-print

Moreover, our lower bound generalizes to

arXiv:1409.2731v1
fatcat:luvihb4my5dnnmegzob5lomsne
*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 ...##
###
Narrow Proofs May Be Maximally Long

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

Moreover, our lower bound generalizes to

doi:10.1109/ccc.2014.36
dblp:conf/coco/AtseriasLN14
fatcat:uek3kmbdobaehdcplxysvsmoaa
*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 ...##
###
Sherali--Adams Strikes Back
[article]

2018
*
arXiv
*
pre-print

We show that

arXiv:1812.09967v1
fatcat:smqmhumghbghvksjejhsje2k7q
*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 ...##
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Limitations of Algebraic Approaches to Graph Isomorphism Testing
[article]

2015
*
arXiv
*
pre-print

We exactly characterise

arXiv:1502.05912v1
fatcat:7ccppsernrhhvfhh77umjkj3yq
*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. ...##
###
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

2018
*
Dagstuhl Reports
*

CSPs constitute a very rich

doi:10.4230/dagrep.8.6.1
dblp:journals/dagstuhl-reports/GroheGZ18
fatcat:3bqo62ly3rgzlnh3bmkvwbuwea
*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. ...##
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The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs
[chapter]

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. ...

##
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Separations in Proof Complexity and TFNP
[article]

2022
*
arXiv
*
pre-print

It is well-known that Resolution

arXiv:2205.02168v2
fatcat:lv4bsib3dza3zb6vu46pmva7ny
*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. ...##
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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
*

By moving to

doi:10.1145/3209108.3209186
dblp:conf/lics/AtseriasO18
fatcat:4pxjokuxwzch5oakuwube2l7j4
*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. ...##
###
Proof Complexity (Dagstuhl Seminar 18051)

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. ...

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

2019
*
arXiv
*
pre-print

Chvatal-Gomory (CG) cuts

arXiv:1709.07966v7
fatcat:ns2h3t4qwzaavc6sgdfpcvyuyu
*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*. ...
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