arXiv subjects

Alex Bortolotti

Publications and source records attributed to Alex Bortolotti.

3 recordsLinked to original sources

arXiv2026

Ideal Membership in Polynomial Calculus: Complexity and Reductions

The Ideal Membership Problem (IMP) asks whether a polynomial f belongs to an ideal of Q[x_1, ..., x_n]. Polynomial Calculus (PC) certifies membership by deriving f from the generators, and a degree-d derivation needs at most n^O(d) steps. We write PC-IMPd for the problem of producing a degree-bounded PC certificate, and call it solvable when one is guaranteed to exist and can be found in time n^O(d). Over Q, unlike over finite fields, a derivation may need exponentially many bits. We study PC-IMPd on instances arising from constraint satisfaction problems, and ask for which constraint languages L it is solvable. Our main contribution is a reduction framework for PC-IMPd, based on pp-definitions, pp-interpretations, and pp-encodings, that mirrors the algebraic approach to CSP complexity. Solvability is preserved by these constructions and, in the language of algebras, by passing to subalgebras, finite direct powers, and homomorphic images. We obtain new tractable classes over ternary and larger domains: every language closed under the median operation on a finite chain has solvable PC-IMPd, by reduction to the Boolean majority algebra, and in particular so does every language over {0, 1, 2} closed under a fixed-value majority. This also places IMPd(L) in P for such languages, advancing the classification of IMPd over ternary domains. In the process, we settle the last open case of the Boolean dichotomy for IMPd(L) and complete the Boolean classification of PC-IMPd(L) with an unconditional lower bound for an instance of PC-IMP1. A recent PC-to-SoS simulation reduces degree-automatability of Sum-of-Squares (the open problem of finding a degree-d SoS proof in time n^O(d) when one exists) to solvability of PC-IMPd. Each new tractable class therefore yields a family of constraint systems on which SoS proofs are degree-automatable.

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arXiv2025

On the Bit Size of Sum-of-Squares Proofs for Symmetric Formulations

The Sum-of-Squares (SoS) hierarchy is a powerful framework for polynomial optimization and proof complexity, offering tight semidefinite relaxations that capture many classical algorithms. Despite its broad applicability, several works have revealed fundamental limitations to SoS automatability. (i) While low-degree SoS proofs are often desirable for tractability, recent works have revealed they may require coefficients of prohibitively large bit size, rendering them computationally infeasible. (ii) Prior works have shown that SoS proofs for seemingly easy problems require high-degree. In particular, this phenomenon also arises in highly symmetric problems. Instances of symmetric problems-particularly those with a small number of constraints-have repeatedly served as benchmarks for establishing high-degree lower bounds in the SoS hierarchy. It has remained unclear whether symmetry can also lead to large bit sizes in SoS proofs, potentially making low-degree proofs computationally infeasible even in symmetric settings. In this work, we resolve this question by proving that symmetry alone does not lead to large bit size SoS proofs. Focusing on symmetric Archimedean instances, we show that low-degree SoS proofs for such systems admit compact, low bit size representations. Together, these results provide a conceptual separation between two sources of SoS hardness-degree and bit size-by showing they do not necessarily align, even in highly symmetric instances. This insight guides future work on automatability and lower bounds: symmetry may necessitate high-degree proofs, but it does not by itself force large coefficients.

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arXiv2025

On the Degree Automatability of Sum-of-Squares Proofs

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.

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