3 Search Results for "Bharathi, Arpitha P."

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.34
On the Degree Automatability of Sum-Of-Squares Proofs

Authors: Alex Bortolotti, Monaldo Mastrolilli, and Luis Felipe Vargas

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are examples of polynomial systems with bounded coefficients that admit low-degree SoS proofs, but these proofs necessarily involve numbers with an exponential number of bits, implying that low-degree SoS proofs cannot always be found efficiently. A sufficient condition derived from the Nullstellensatz proof system [Raghavendra and Weitz (2017)] identifies cases where bit complexity issues can be circumvented. One of the main problems left open by Raghavendra and Weitz is proving any result for refutations, as their condition applies only to polynomial systems with a large set of solutions. In this work, we broaden the class of polynomial systems for which degree-d SoS proofs can be automated. To achieve this, we develop a new criterion and we demonstrate how our criterion applies to polynomial systems beyond the scope of Raghavendra and Weitz’s result. In particular, we establish a separation for instances arising from Constraint Satisfaction Problems (CSPs). Moreover, our result extends to refutations, establishing that polynomial-time refutation is possible for broad classes of polynomial time solvable constraint problems, highlighting a first advancement in this area.
Cite as

Alex Bortolotti, Monaldo Mastrolilli, and Luis Felipe Vargas. On the Degree Automatability of Sum-Of-Squares Proofs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bortolotti_et_al:LIPIcs.ICALP.2025.34,
  author =	{Bortolotti, Alex and Mastrolilli, Monaldo and Vargas, Luis Felipe},
  title =	{{On the Degree Automatability of Sum-Of-Squares Proofs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{34:1--34:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.34},
  URN =		{urn:nbn:de:0030-drops-234110},
  doi =		{10.4230/LIPIcs.ICALP.2025.34},
  annote =	{Keywords: Sum of squares, Polynomial calculus, Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems, Proof complexity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.16
Ideal Membership Problem for Boolean Minority and Dual Discriminator

Authors: Arpitha P. Bharathi and Monaldo Mastrolilli

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.13
Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain

Authors: Arpitha P. Bharathi and Monaldo Mastrolilli

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

Abstract
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.
Cite as

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)

Copy BibTex To Clipboard

@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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