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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

The polynomial Ideal Membership Problem (IMP) tests if an input polynomial f ∈ 𝔽[x_1,… ,x_n] with coefficients from a field 𝔽 belongs to a given ideal I ⊆ 𝔽[x_1,… ,x_n]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMP_d).
A dichotomy result between "hard" (NP-hard) and "easy" (polynomial time) IMPs was achieved for Constraint Satisfaction Problems over finite domains [Andrei A. Bulatov, 2017; Dmitriy Zhuk, 2020] (this is equivalent to IMP_0) and IMP_d for the Boolean domain [Mastrolilli, 2019], both based on the classification of the IMP through functions called polymorphisms. For the latter result, there are only six polymorphisms to be studied in order to achieve a full dichotomy result for the IMP_d. The complexity of the IMP_d for five of these polymorphisms has been solved in [Mastrolilli, 2019] whereas for the ternary minority polymorphism it was incorrectly declared in [Mastrolilli, 2019] to have been resolved by a previous result. In this paper we provide the missing link by proving that the IMP_d for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the precise borderline of tractability for the IMP_d for constrained problems over the Boolean domain. We also prove that the proof of membership for the IMP_d for problems constrained by the dual discriminator polymorphism over any finite domain can also be found in polynomial time. Bulatov and Rafiey [Andrei A. Bulatov and Akbar Rafiey, 2020] recently proved that the IMP_d for this polymorphism is decidable in polynomial time, without needing a proof of membership. Our result gives a proof of membership and can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.

Arpitha P. Bharathi and Monaldo Mastrolilli. Ideal Membership Problem for Boolean Minority and Dual Discriminator. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bharathi_et_al:LIPIcs.MFCS.2021.16, author = {Bharathi, Arpitha P. and Mastrolilli, Monaldo}, title = {{Ideal Membership Problem for Boolean Minority and Dual Discriminator}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {16:1--16:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.16}, URN = {urn:nbn:de:0030-drops-144560}, doi = {10.4230/LIPIcs.MFCS.2021.16}, annote = {Keywords: Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

The Ideal Membership Problem (IMP) asks if an input polynomial f ∈ 𝔽[x₁,… ,x_n] with coefficients from a field 𝔽 belongs to an input ideal I ⊆ 𝔽[x₁,… ,x_n]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMP_d). Our main interest is in understanding when the inherent combinatorial structure of the ideals makes the IMP_d "hard" (NP-hard) or "easy" (polynomial time) to solve.
Such a dichotomy result between "hard" and "easy" IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Andrei A. Bulatov, 2017; Dmitriy Zhuk, 2017] (this is equivalent to IMP₀) and IMP_d for the Boolean domain [Mastrolilli, 2019], both based on the classification of the IMP through functions called polymorphisms. For the latter result, each polymorphism determined the complexity of the computation of a suitable Gröbner basis.
In this paper we consider a 3-element domain and a majority polymorphism (constraints under this polymorphism are a generalisation of the 2-SAT problem). By using properties of the majority polymorphism and assuming graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under the majority polymorphism can be computed in polynomial time. This proves polynomial time solvability of the IMP_d for these constrained problems. We conjecture that this result can be extended to a general finite domain of size k = O(1). This is a first step towards the long term and challenging goal of generalizing the dichotomy results of solvability of the IMP_d for a finite domain.

Arpitha P. Bharathi and Monaldo Mastrolilli. Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bharathi_et_al:LIPIcs.MFCS.2020.13, author = {Bharathi, Arpitha P. and Mastrolilli, Monaldo}, title = {{Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.13}, URN = {urn:nbn:de:0030-drops-126829}, doi = {10.4230/LIPIcs.MFCS.2020.13}, annote = {Keywords: Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We give two results concerning the power of the Sum-Of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that (n+2d-1)/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.

Adam Kurpisz, Samuli Leppänen, and Monaldo Mastrolilli. Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 78:1-78:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kurpisz_et_al:LIPIcs.ICALP.2016.78, author = {Kurpisz, Adam and Lepp\"{a}nen, Samuli and Mastrolilli, Monaldo}, title = {{Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {78:1--78:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.78}, URN = {urn:nbn:de:0030-drops-63368}, doi = {10.4230/LIPIcs.ICALP.2016.78}, annote = {Keywords: SoS/Lasserre hierarchy, lift and project methods, binary polynomial optimization} }