arXiv subjects

Albert Atserias

Publications and source records attributed to Albert Atserias.

At least 19 recordsLinked to original sources

arXiv2026

The Proof Analysis Problem

Atserias and Müller (JACM, 2020) proved that for every unsatisfiable CNF formula $φ$, the formula $\operatorname{Ref}(φ)$, stating "$φ$ has small Resolution refutations", does not have subexponential-size Resolution refutations. Conversely, when $φ$ is satisfiable, Pudlák (TCS, 2003) showed how to construct a polynomial-size Resolution refutation of $\operatorname{Ref}(φ)$ given a satisfying assignment of $φ$. A question that remained open is: do all short Resolution refutations of $\operatorname{Ref}(φ)$ explicitly leak a satisfying assignment of $φ$? We answer this question affirmatively by giving a polynomial-time algorithm that extracts a satisfying assignment for $φ$ given any short Resolution refutation of $\operatorname{Ref}(φ)$. The algorithm follows from a new feasibly constructive proof of the Atserias-Müller lower bound, formalizable in Cook's theory $\mathsf{PV_1}$ of bounded arithmetic. Motivated by this, we introduce a computational problem concerning Resolution lower bounds: the Proof Analysis Problem (PAP). For a proof system $Q$, the Proof Analysis Problem for $Q$ asks, given a CNF formula $φ$ and a $Q$-proof of a Resolution lower bound for $φ$, encoded as $\neg \operatorname{Ref}(φ)$, whether $φ$ is satisfiable. In contrast to PAP for Resolution, we prove that PAP for Extended Frege (EF) is NP-complete. Our results yield new insights into proof complexity: (i) every proof system simulating EF is (weakly) automatable if and only if it is (weakly) automatable on formulas stating Resolution lower bounds; (ii) we provide Ref formulas exponentially hard for bounded-depth Frege systems; and (iii) for every strong enough theory of arithmetic $T$ we construct unsatisfiable CNF formulas exponentially hard for Resolution but for which $T$ cannot prove even a quadratic lower bound.

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arXiv2026

From Gödel incompleteness to the consistency of circuit lower bounds

We prove that the bounded arithmetic theory $S^1_2$ is consistent with EXP $\not\subseteq$ P/poly. More generally, we show that certain separations of $V^1_2$ from a theory $T$ imply the consistency of $T$ with EXP $\not\subseteq$ P/poly. For $T=S^1_2$, Takeuti (1988) established such a separation using a variant of Gödel's consistency statement. Analogous results hold for PSPACE $\not\subseteq$ P/poly but the required separations of theories are yet unknown. Finally, we give magnification results for the hardness of proving almost-everywhere versions of these lower bounds.

math.LO↗

arXiv2026

Hard Clique Formulas for Resolution

We show how to convert any unsatisfiable 3-CNF formula which is sparse and exponentially hard to refute in Resolution into a negative instance of the $k$-clique problem whose corresponding natural encoding as a CNF formula is $n^{Ω(k)}$-hard to refute in Resolution. This applies to any function $k = k(n)$ of the number $n$ of vertices, provided $k_0 \leq k \leq n^{1/c_0}$, where $k_0$ and $c_0$ are small constants. We establish this by demonstrating that Resolution can simulate the correctness proof of a particular kind of reduction from 3-SAT to the parameterized clique problem. This also re-establishes the known conditional hardness result for $k$-clique which states that if the Exponential Time Hypothesis (ETH) holds, then the $k$-clique problem cannot be solved in time $n^{o(k)}$. Since it is known that the analogue of ETH holds for Resolution, unconditionally and with explicit hard instances, this gives a way to obtain explicit instances of $k$-clique that are unconditionally $n^{Ω(k)}$-hard to refute in Resolution. This solves an open problem that appeared published in the literature at least twice.

cs.CC↗

arXiv2025

Gamma Acyclicity, Annotated Relations, and Consistency Witness Functions

During the early days of relational database theory it was realized that "acyclic" database schemas possess a number of desirable semantic properties. In fact, three different notions of "acyclicity" were identified and extensively investigated during the 1980s, namely, alpha-acyclicity, beta-acyclicity, and gamma-acyclicity. Much more recently, the study of alpha-acyclicity was extended to annotated relations, where the annotations are values from some positive commutative monoid. The recent results about alpha-acyclic schemas and annotated relations give rise to results about beta-acyclic schemas and annotated relations, since a schema is beta-acyclic if and only if every sub-schema of it is alpha-acyclic. Here, we study gamma-acyclic schemas and annotated relations. Our main finding is that the desirable semantic properties of gamma-acyclic schemas extend to annotated relations, provided the annotations come from a positive commutative monoid that has the transportation property. Furthermore, the results reported here shed light on the role of the join of two standard relations, Specifically, our results reveal that the only relevant property of the join of two standard relations is that it is a witness to the consistency of the two relations, provided that these two relations are consistent. For the more abstract setting of annotated relations, this property of the standard join is captured by the notion of a consistency witness function, a notion which we systematically investigate in this work.

cs.DB↗

arXiv2025

Simple general magnification of circuit lower bounds

We introduce a technically and conceptually simple approach to magnification of circuit and formula lower bounds. Central to the method are so-called distinguishers, sparse matrices that retain some of the key properties of error-correcting codes. As applications, we generalize and strengthen known general (not problem specific) magnification results and in particular achieve magnification thresholds below known lower bounds. For example, we show that fixed-polynomial formula-size lower bounds for NP are implied by slightly superlinear formula-size lower bounds for approximating any sufficiently sparse problem in NP. We also show that the thresholds achieved are sharp. Additionally, our approach yields a uniform magnification result for the Minimum Circuit Size Problem (MCSP). This seems to sidestep the localization barrier.

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arXiv2024

Feasibly Constructive Proof of Schwartz-Zippel Lemma and the Complexity of Finding Hitting Sets

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, and combinatorics. We give a new proof of the lemma that offers some advantages over the standard proof. First, the new proof is more constructive than previously known proofs. For every given side-length of the cube, the proof constructs a polynomial-time computable and polynomial-time invertible surjection onto the set of roots in the cube. The domain of the surjection is tight, thus showing that the set of roots on the cube can be compressed. Second, the new proof can be formalised in Buss' bounded arithmetic theory $\mathrm{S}^1_2$ for polynomial-time reasoning. One consequence of this is that the theory $\mathrm{S}^1_2 + \mathrm{dWPHP(PV)}$ for approximate counting can prove that the problem of verifying polynomial identities (PIT) can be solved by polynomial-size circuits. The same theory can also prove the existence of small hitting sets for any explicitly described class of polynomials of polynomial degree. To complete the picture we show that the existence of such hitting sets is \emph{equivalent} to the surjective weak pigeonhole principle $\mathrm{dWPHP(PV)}$, over the theory $\mathrm{S}^1_2$. This is a contribution to a line of research studying the reverse mathematics of computational complexity. One consequence of this is that the problem of constructing small hitting sets for such classes is complete for the class APEPP of explicit construction problems whose totality follows from the probabilistic method. This class is also known and studied as the class of Range Avoidance Problems.

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arXiv2023

Consistency of Relations over Monoids

The interplay between local consistency and global consistency has been the object of study in several different areas, including probability theory, relational databases, and quantum information. For relational databases, Beeri, Fagin, Maier, and Yannakakis showed that a database schema is acyclic if and only if it has the local-to-global consistency property for relations, which means that every collection of pairwise consistent relations over the schema is globally consistent. More recently, the same result has been shown under bag semantics. In this paper, we carry out a systematic study of local vs.\ global consistency for relations over positive commutative monoids, which is a common generalization of ordinary relations and bags. Let $\mathbb K$ be an arbitrary positive commutative monoid. We begin by showing that acyclicity of the schema is a necessary condition for the local-to-global consistency property for $\mathbb K$-relations to hold. Unlike the case of ordinary relations and bags, however, we show that acyclicity is not always sufficient. After this, we characterize the positive commutative monoids for which acyclicity is both necessary and sufficient for the local-to-global consistency property to hold; this characterization involves a combinatorial property of monoids, which we call the \emph{transportation property}. We then identify several different classes of monoids that possess the transportation property. As our final contribution, we introduce a modified notion of local consistency of $\mathbb{K}$-relations, which we call \emph{pairwise consistency up to the free cover}. We prove that, for all positive commutative monoids $\mathbb{K}$, even those without the transportation property, acyclicity is both necessary and sufficient for every family of $\mathbb{K}$-relations that is pairwise consistent up to the free cover to be globally consistent.

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arXiv2023

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Graph Isomorphism Problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. 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 a small loss in the degree.

cs.LO↗

arXiv2023

On the Consistency of Circuit Lower Bounds for Non-Deterministic Time

We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V$^0_2$ is consistent with the conjecture that NEXP $\not\subseteq$ P/poly, i.e., some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. The same techniques establish the same results with NEXP replaced by the class of problems that are decidable in non-deterministic barely superpolynomial time such as NTIME$(n^{O(\log\log\log n)})$. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds.

cs.CC↗

arXiv2023

Circular (Yet Sound) Proofs

Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type of structural restriction for proofs in rule-based proof systems. In this definition, proofs are directed graphs of derived formulas in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs "circular". We show that, for all sets of standard inference rules with single or multiple conclusions, circular proofs are sound. We start the study of the proof complexity of circular proofs at Circular Resolution, the circular version of Resolution. We immediately see that Circular Resolution is stronger than Dag-like Resolution since, as we show, the propositional encoding of the pigeonhole principle has circular Resolution proofs of polynomial size. Furthermore, for derivations of clauses from clauses, we show that Circular Resolution is, surprisingly, equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: 1) polynomial-time (LP-based) algorithms that find Circular Resolution proofs of constant width, 2) examples that separate Circular from Dag-like Resolution, such as the pigeonhole principle and its variants, and 3) exponentially hard cases for Circular Resolution. Contrary to the case of Circular Resolution, for Frege we show that circular proofs can be converted into tree-like proofs with at most polynomial overhead.

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arXiv2021

Promise Constraint Satisfaction and Width

We study the power of the bounded-width consistency algorithm in the context of the fixed-template Promise Constraint Satisfaction Problem (PCSP). Our main technical finding is that the template of every PCSP that is solvable in bounded width satisfies a certain structural condition implying that its algebraic closure-properties include weak near unanimity polymorphisms of all large arities. While this parallels the standard (non-promise) CSP theory, the method of proof is quite different and applies even to the regime of sublinear width. We also show that, in contrast with the CSP world, the presence of weak near unanimity polymorphisms of all large arities does not guarantee solvability in bounded width. The separating example is even solvable in the second level of the Sherali-Adams (SA) hierarchy of linear programming relaxations. This shows that, unlike for CSPs, linear programming can be stronger than bounded width. A direct application of these methods also show that the problem of $q$-coloring $p$-colorable graphs is not solvable in bounded or even sublinear width, for any two constants $p$ and $q$ such that $3 \leq p \leq q$. Turning to algorithms, we note that Wigderson's algorithm for $O(\sqrt{n})$-coloring $3$-colorable graphs with $n$ vertices is implementable in width $4$. Indeed, by generalizing the method we see that, for any $ε> 0$ smaller than $1/2$, the optimal width for solving the problem of $O(n^ε)$-coloring $3$-colorable graphs with $n$ vertices lies between $n^{1-3ε}$ and $n^{1-2ε}$. The upper bound gives a simple $2^{Θ(n^{1-2ε}\log(n))}$-time algorithm that, asymptotically, beats the straightforward $2^{Θ(n^{1-ε})}$ bound that follows from partitioning the graph into $O(n^ε)$ many independent parts each of size $O(n^{1-ε})$.

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arXiv2021

On the Expressive Power of Homomorphism Counts

A classical result by Lovász asserts that two graphs $G$ and $H$ are isomorphic if and only if they have the same left profile, that is, for every graph $F$, the number of homomorphisms from $F$ to $G$ coincides with the number of homomorphisms from $F$ to $H$. Dvor{á}k and later on Dell, Grohe, and Rattan showed that restrictions of the left profile to a class of graphs can capture several different relaxations of isomorphism, including equivalence in counting logics with a fixed number of variables (which contains fractional isomorphism as a special case) and co-spectrality (i.e., two graphs having the same characteristic polynomial). On the other side, a result by Chaudhuri and Vardi asserts that isomorphism is also captured by the right profile, that is, two graphs $G$ and $H$ are isomorphic if and only if for every graph $F$, the number of homomorphisms from $G$ to $F$ coincides with the number of homomorphisms from $H$ to $F$. In this paper, we embark on a study of the restrictions of the right profile by investigating relaxations of isomorphism that can or cannot be captured by restricting the right profile to a fixed class of graphs. Our results unveil striking differences between the expressive power of the left profile and the right profile. We show that fractional isomorphism, equivalence in counting logics with a fixed number of variables, and co-spectrality cannot be captured by restricting the right profile to a class of graphs. In the opposite direction, we show that chromatic equivalence cannot be captured by restricting the left profile to a class of graphs, while, clearly, it can be captured by restricting the right profile to the class of all cliques.

math.CO↗

arXiv2020

Structure and Complexity of Bag Consistency

Since the early days of relational databases, it was realized that acyclic hypergraphs give rise to database schemas with desirable structural and algorithmic properties. In a by-now classical paper, Beeri, Fagin, Maier, and Yannakakis established several different equivalent characterizations of acyclicity; in particular, they showed that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for relations over that schema holds, which means that every collection of pairwise consistent relations over the schema is globally consistent. Even though real-life databases consist of bags (multisets), there has not been a study of the interplay between local consistency and global consistency for bags. We embark on such a study here and we first show that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to global consistency property for bags over that schema holds. After this, we explore algorithmic aspects of global consistency for bags by analyzing the computational complexity of the global consistency problem for bags: given a collection of bags, are these bags globally consistent? We show that this problem is in NP, even when the schema is part of the input. We then establish the following dichotomy theorem for fixed schemas: if the schema is acyclic, then the global consistency problem for bags is solvable in polynomial time, while if the schema is cyclic, then the global consistency problem for bags is NP-complete. The latter result contrasts sharply with the state of affairs for relations, where, for each fixed schema, the global consistency problem for relations is solvable in polynomial time.

cs.DB↗

arXiv2020

Clique Is Hard on Average for Regular Resolution

We prove that for $k \ll \sqrt[4]{n}$ regular resolution requires length $n^{Ω(k)}$ to establish that an Erdős-Rényi graph with appropriately chosen edge density does not contain a $k$-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional $n^{Ω(k)}$ lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

cs.CC↗

arXiv2020

Consistency, Acyclicity, and Positive Semirings

In several different settings, one comes across situations in which the objects of study are locally consistent but globally inconsistent. Earlier work about probability distributions by Vorob'ev (1962) and about database relations by Beeri, Fagin, Maier, Yannakakis (1983) produced characterizations of when local consistency always implies global consistency. Towards a common generalization of these results, we consider K-relations, that is, relations over a set of attributes such that each tuple in the relation is associated with an element from an arbitrary, but fixed, positive semiring K. We introduce the notions of projection of a K-relation, consistency of two K-relations, and global consistency of a collection of K-relations; these notions are natural extensions of the corresponding notions about probability distributions and database relations. We then show that a collection of sets of attributes has the property that every pairwise consistent collection of K-relations over those attributes is globally consistent if and only if the sets of attributes form an acyclic hypergraph. This generalizes the aforementioned results by Vorob'ev and by Beeri et al., and demonstrates that K-relations over positive semirings constitute a natural framework for the study of the interplay between local and global consistency. In the course of the proof, we introduce a notion of join of two K-relations and argue that it is the "right" generalization of the join of two database relations. Furthermore, to show that non-acyclic hypergraphs yield pairwise consistent K-relations that are globally inconsistent, we generalize a construction by Tseitin (1968) in his study of hard-to-prove tautologies in propositional logic.

cs.DB↗

arXiv2019

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively.

cs.CC↗

arXiv2019

Definable Inapproximability: New Challenges for Duplicator

We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.

cs.LO↗

arXiv2019

Non-Homogenizable Classes of Finite Structures

Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability.

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