Abstract. The Sum-of-Squares ([Formula: see text]) 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 [Formula: see text] automatability. (i) While low-degree [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] hierarchy. It has remained unclear whether symmetry can also lead to large bit sizes in [Formula: see text] 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 [Formula: see text] proofs. Focusing on symmetric Archimedean instances, we show that low-degree [Formula: see text] proofs for such systems admit compact, low bit size representations. Together, these results provide a conceptual separation between two sources of [Formula: see text] 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.

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.

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B\&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025

Abstract. We consider the polynomial ideal membership problem (IMP) for ideals encoding combinatorial problems that are instances of constraint satisfaction problems over a finite language. In this paper, the input polynomial [Formula: see text] has degree at most [Formula: see text] (we call this problem IMP[Formula: see text]). We bridge the gap in [M. Mastrolilli, The complexity of the ideal membership problem for constrained problems over the Boolean domain, in SODA ’19, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, PA, Society for Industrial and Applied Mathematics, 2019, pp. 456–475] by proving that the IMP[Formula: see text] for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean [Formula: see text]. We also prove that the proof of membership for the [Formula: see text] for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and sum-of-squares proofs.

In this paper we examine polynomial ideals that are the vanishing ideals of solution sets of combinatorial problems encoded by constraint satisfaction problems over a finite language. We consider a 3-element domain and the dual discriminator polymorphism (constraints under this polymorphism are a generalization of the 2-satisfiability problem). Assuming the graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under this polymorphism can be computed in polynomial time. This proves polynomial time solvability of the ideal membership problem (IMP) with restrictions on degree $d=O(1)$, which we call IMP$_d$, for these constrained problems. It is a first step toward the challenging long-term goal of identifying when IMP$_d$ is polynomial time solvable for a finite domain.

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.

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.

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.

Given an ideal I and a polynomial f the Ideal Membership Problem is to test if f ∊ I. This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the Ideal Membership Problem for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the precise borderline of tractability. By using Gröbner bases techniques, we generalize Schaefer's dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems over the Boolean domain to be either in P or NP-hard. This paper is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares proofs [O'Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the Ideal Membership Problem tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.

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.

We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a k-hypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budget-constrained auctions and generalizes well-studied problems in combinatorial optimization: for example the weighted (unweighted) k-hypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1. Let ε > 0 denote an arbitrarily small constant. In this paper we obtain the following main results: We give a randomized (k + 1 + ∊) approximation algorithm for HAP, which is based on rounding the 1-round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted k-hypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA'10]. For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O(√k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + ∊)/2 approximation ratio by Berman [SWAT'00] for the (less general) weighted k-hypergraph matching problem. For the unweighted k-hypergraph matching problem, we present a (k + 1 + ∊)/3 approximation in quasipolynomial time. This improves over the (k + 2)/3 approximation by Halldórsson [SODA'95] (also in quasipolynomial time). In particular this suggests that a 4/3 + ∊ approximation for 3-dimensional matching might exist, whereas the currently best known polynomial-time approximation ratio is 3/2.

We consider several variants of the job shop problem that is a fundamental and classical problem in scheduling. The currently best approximation algorithms have worse than logarithmic performance guarantee, but the only previously known inapproximability result says that it is NP-hard to approximate job shops within a factor less than 5/4. Closing this big approximability gap is a well-known and long-standing open problem. This article closes many gaps in our understanding of the hardness of this problem and answers several open questions in the literature. It is shown the first nonconstant inapproximability result that almost matches the best-known approximation algorithm for acyclic job shops. The same bounds hold for the general version of flow shops, where jobs are not required to be processed on each machine. Similar inapproximability results are obtained when the objective is to minimize the sum of completion times. It is also shown that the problem with two machines and the preemptive variant with three machines have no PTAS.

We consider the Minimum Linear Arrangement problem and the (Uniform) Sparsest Cut problem. So far, these two notorious NP-hard graph problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme, unless NP-complete problems can be solved in randomized subexponential time. Furthermore, we show that the same techniques can be used for the Maximum Edge Biclique problem, for which we obtain a hardness factor similar to previous results but under a more standard assumption.

For every euro > 0, we show that the (acyclic) job shop problem cannot be approximated within ratio O(log1+eurolb), unless NP has quasi-polynomial Las-Vegas algorithms, and where lb denotes a trivial lower bound on the optimal value. This almost matches the best known results for acyclic job shops, since an O(log1+eurolb)-approximate solution can be obtained in polynomial time for every euro > 0. Recently, a PTAS was given for the job shop problem, where the number of machines and the number of operations per job are assumed to be constant. Under P ne NP, and when the number mu of operations per job is a constant, we provide an inapproximability result whose value grows with mu to infinity. Moreover, we show that the problem with two machines and the preemptive variant with three machines have no PTAS, unless NP has quasi-polynomial algorithms. These results show that the restrictions on the number of machines and operations per job are necessary to obtain a PTAS.In summary, the presented results close many gaps in our understanding of the hardness of the job shop problem and resolve (negatively) several open problems in the literature.

We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.