arXiv subjects

Gal Beniamini

Publications and source records attributed to Gal Beniamini.

10 recordsLinked to original sources

arXiv2025

FormulaOne: Measuring the Depth of Algorithmic Reasoning Beyond Competitive Programming

Frontier AI models demonstrate formidable breadth of knowledge. But how close are they to true human -- or superhuman -- expertise? Genuine experts can tackle the hardest problems and push the boundaries of scientific understanding. To illuminate the limits of frontier model capabilities, we turn away from contrived competitive programming puzzles, and instead focus on real-life research problems. We construct FormulaOne, a benchmark that lies at the intersection of graph theory, logic, and algorithms, all well within the training distribution of frontier models. Our problems are incredibly demanding, requiring an array of reasoning steps. The dataset has three key properties. First, it is of commercial interest and relates to practical large-scale optimisation problems, such as those arising in routing, scheduling, and network design. Second, it is generated from the highly expressive framework of Monadic Second-Order (MSO) logic on graphs, paving the way toward automatic problem generation at scale; ideal for building RL environments. Third, many of our problems are intimately related to the frontier of theoretical computer science, and to central conjectures therein, such as the Strong Exponential Time Hypothesis (SETH). As such, any significant algorithmic progress on our dataset, beyond known results, could carry profound theoretical implications. Remarkably, state-of-the-art models like OpenAI's o3 fail entirely on FormulaOne, solving less than 1% of the questions, even when given 10 attempts and explanatory fewshot examples -- highlighting how far they remain from expert-level understanding in some domains. To support further research, we additionally curate FormulaOne-Warmup, offering a set of simpler tasks, from the same distribution. We release the full corpus along with a comprehensive evaluation framework.

cs.AI↗

arXiv2024

Artificial Expert Intelligence through PAC-reasoning

Artificial Expert Intelligence (AEI) seeks to transcend the limitations of both Artificial General Intelligence (AGI) and narrow AI by integrating domain-specific expertise with critical, precise reasoning capabilities akin to those of top human experts. Existing AI systems often excel at predefined tasks but struggle with adaptability and precision in novel problem-solving. To overcome this, AEI introduces a framework for ``Probably Approximately Correct (PAC) Reasoning". This paradigm provides robust theoretical guarantees for reliably decomposing complex problems, with a practical mechanism for controlling reasoning precision. In reference to the division of human thought into System 1 for intuitive thinking and System 2 for reflective reasoning~\citep{tversky1974judgment}, we refer to this new type of reasoning as System 3 for precise reasoning, inspired by the rigor of the scientific method. AEI thus establishes a foundation for error-bounded, inference-time learning.

cs.AI↗

arXiv2024

Counting Permutation Patterns with Multidimensional Trees

We consider the well-studied pattern counting problem: given a permutation $π\in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $τ\in \mathbb{S}_k$ in $π$. Our first result is an $\widetilde{\mathcal{O}}(n^2)$-time algorithm for $k=6$ and $k=7$. The proof relies heavily on a new family of graphs that we introduce, called pattern-trees. Every such tree corresponds to an integer linear combination of permutations in $\mathbb{S}_k$, and is associated with linear extensions of partially ordered sets. We design an evaluation algorithm for these combinations, and apply it to a family of linearly-independent trees. For $k=8$, we show a barrier: the subspace spanned by trees in the previous family has dimension exactly $|\mathbb{S}_8| - 1$, one less than required. Our second result is an $\widetilde{\mathcal{O}}(n^{7/4})$-time algorithm for $k=5$. This algorithm extends the framework of pattern-trees by speeding-up their evaluation in certain cases. A key component of the proof is the introduction of pair-rectangle-trees, a data structure for dominance counting.

cs.DS↗

arXiv2024

The Rank-Ramsey Problem and the Log-Rank Conjecture

A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erd\H{o}s-R\'enyi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.

math.CO↗

arXiv2023

How Balanced Can Permutations Be?

A permutation $π\in \mathbb{S}_n$ is $k$-balanced if every permutation of order $k$ occurs in $π$ equally often, through order-isomorphism. In this paper, we explicitly construct $k$-balanced permutations for $k \le 3$, and every $n$ that satisfies the necessary divisibility conditions. In contrast, we prove that for $k \ge 4$, no such permutations exist. In fact, we show that in the case $k \ge 4$, every $n$-element permutation is at least $Ω_n(n^{k-1})$ far from being $k$-balanced. This lower bound is matched for $k=4$, by a construction based on the Erdős-Szekeres permutation.

math.CO↗

arXiv2022

The $\mathbb{F}_2$-Rank and Size of Graphs

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.

math.CO↗

arXiv2022

Algebraic Representations of Unique Bipartite Perfect Matching

We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results, we show that, surprisingly, the dual description is sparse and has low $\ell_1$-norm -- only exponential in $Θ(n \log n)$, and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula. These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight $Θ(n \log n)$ bound on the log-rank of the associated two-party communication task.

cs.DM↗

arXiv2022

The Approximate Degree of Bipartite Perfect Matching

The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the $\ell_\infty$-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching, is $\widetildeΘ(n^{3/2})$. The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small $\ell_1$-norm -- only exponential in $Θ(n \log n)$. The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube \cite{aaronson2020degree, huang2019induced}. We show that the spectral sensitivity of perfect matching is exactly $Θ(n^{3/2})$.

cs.DM↗

arXiv2020

Sparsifying the Operators of Fast Matrix Multiplication Algorithms

Fast matrix multiplication algorithms may be useful, provided that their running time is good in practice. Particularly, the leading coefficient of their arithmetic complexity needs to be small. Many sub-cubic algorithms have large leading coefficients, rendering them impractical. Karstadt and Schwartz (SPAA'17, JACM'20) demonstrated how to reduce these coefficients by sparsifying an algorithm's bilinear operator. Unfortunately, the problem of finding optimal sparsifications is NP-Hard. We obtain three new methods to this end, and apply them to existing fast matrix multiplication algorithms, thus improving their leading coefficients. These methods have an exponential worst case running time, but run fast in practice and improve the performance of many fast matrix multiplication algorithms. Two of the methods are guaranteed to produce leading coefficients that, under some assumptions, are optimal.

cs.DS↗

arXiv2020

Bipartite Perfect Matching as a Real Polynomial

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in $Θ(n \log n)$. Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered" is Eulerian.

cs.DM↗