arXiv subjects

Pavel Arkhipov

Publications and source records attributed to Pavel Arkhipov.

9 recordsLinked to original sources

arXiv2026

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

cs.LG↗

arXiv2026

Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm

Minimum weight perfect matching is a fundamental problem in combinatorial optimization. Since 2009, Blossom V has been the leading practical implementation. We present Blossom VI, a practical algorithm that improves upon Blossom V. We test the performance of our implementation on nine benchmark families with graphs containing up to roughly 10 million edges. Blossom VI is significantly faster on the hardest tested instances, while being at most a factor of two slower on families where Blossom V already scales nearly linearly. Blossom VI is a primal-dual solver whose primal phase computes a maximum-cardinality matching in the zero-slack subgraph using cherry trees. At the end of each primal phase, Blossom VI contracts entire cherry blossoms instead of sequences of nested traditional blossoms that would be created by Blossom V. This produces shallower contraction hierarchies and avoids costly cascades of blossom expansions. Our measurements show that reduced blossom depth and fewer expensive expansion operations can explain the observed improvement.

cs.DS↗

arXiv2026

Improved Error Bounds for Pure Differentially Private Continual Counting via Matrix Factorization

Continual counting under pure differential privacy is one of the simplest and most well-studied problems in the continual observation model. Nevertheless, an asymptotic gap remains between the best known upper and lower bounds for maximum squared error and mean squared error: the upper bound is $O(ε^{-2}\log^3 n)$, while the lower bound is $Ω(ε^{-2}\log^2 n)$, for both error metrics. The best known constant in the upper bound is achieved by the $k$-ary tree mechanism with the subtraction trick, due to Andersson, Pagh, Steiner, and Torkamani (FORC 2025). In this work, we improve the leading constant in the maximum squared error and the mean squared error. Our approach uses a general matrix factorization mechanism, yielding an improved bound for pure-DP continual counting that does not rely on a tree-based construction. The mechanism starts from a good-quality low-dimensional factorization, obtained via gradient-based optimization, and gives an explicit matrix construction that lifts this factorization to arbitrarily large dimensions, further improving its error guarantees. We offer an efficient algorithmic implementation of our mechanism. On the lower-bound side, we prove an $Ω(ε^{-2}\log^3 n)$ lower bound for the class of factorizations whose matrices have entries in $\{0,1\}$, matching the upper-bound asymptotics for this class. This class includes the binary tree mechanism and $k$-ary tree mechanisms without the subtraction trick. Extending this lower bound to arbitrary matrix factorizations, and beyond the matrix mechanism altogether, remains an open problem.

cs.DS↗

arXiv2026

Tighter relaxations for MAP-MRF optimization via Singleton Arc Consistency

We consider the MAP-MRF inference task, that is, minimizing a function of discrete variables represented as a sum of unary and pairwise terms. A prominent approach for tackling this NP-hard problem in practice is to solve its natural LP relaxation and then iteratively tighten the relaxation by adding clusters. Based on some theoretical observations, we propose a new technique for identifying such clusters. It works by running the Singleton Arc Consistency algorithm in a certain CSP instance. Experimental results indicate that the new tightening technique outperforms the previous approach by [Sontag et al. UAI 2012] that searches for frustrated cycles. Our code will be made available at https://github.com/vnk-ist/MAP-MRF/.

cs.DS↗

arXiv2026

Greedy matroid base packings with applications to dynamic graph density and orientations

Greedy minimum weight spanning tree packings have proven to be useful in connectivity-related problems. We study the process of greedy minimum weight base packings in general matroids and explore its applications. For general matroids, we observe two characterizations of the limit of the base packings (``the vector of ideal loads''). Specialized to graphic matroids, it implies the characterizations from [Cen, Fleischmann, Li, Li, Panigrahi, FOCS'25], namely, their entropy-minimization theorem and their bottom-up cut hierarchy. We give combinatorial results on the greedy tree packings. We show that a tree packing of $O(λ^5\log m)$ trees contains a tree crossing some min-cut once, which improves the bound $O(λ^7\log^3m)$ from [Thorup, Combinatorica'07]. We also strengthen the lower bound on the edge load convergence rate from [de Vos, Christiansen, SODA'25], showing that Thorup's upper bound is tight up to a logarithmic factor. When specialized to bicircular matroids, our results yield an algorithm for the approximate fully-dynamic densest subgraph density $ρ$. We maintain a $(1+\varepsilon)$-approximation of the density with a worst-case update time $O((ρ_{\max}\varepsilon^{-2}+\varepsilon^{-4})ρ_{\max}\log^3 m)$, where $ρ_{\max}$ is a fixed known upper bound on $ρ$. This complexity is worse than the state-of-the-art dynamic approximate density. However, our algorithm offers a new approach to the problem, which could be appealing due to its simplicity. We also can maintain an implicit fractional out-orientation with a guarantee that all out-degrees are at most $(1+\varepsilon)ρ$. Our algorithms above work by greedily packing pseudoforests, and require maintenance of a minimum-weight pseudoforest in a dynamically changing graph. We show that this problem can be solved in $O(\log n)$ worst-case time per edge insertion or deletion.

cs.DS↗

arXiv2026

Faster algorithms for packing forests in graphs and related problems

We consider several problems related to packing forests in graphs. The first one is to find $k$ edge-disjoint forests in a directed graph $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We describe a min-max characterization for this problem and show that it can be solved in almost linear time for fixed $k$, extending the algorithm of [Gabow, 1995]. Specifically, the complexity is $O(k δm \log n)$, where $n, m$ are the number of vertices and edges in $G$ respectively, and $δ= \max\{1, k - k_G\}$, where $k_G$ is the edge connectivity of the graph. Using our solution to this problem, we improve complexities for two existing applications: (1) $k$-forest problem: find $k$ forests in an undirected graph $G$ maximizing the number of edges in their union. We show how to solve this problem in $O(k^3 \min\{kn, m\} \log^2 n + k \cdot{\rm MAXFLOW}(m, m) \log n)$ time, breaking the $O_k(n^{3/2})$ complexity barrier of previously known approaches. (2) Directed edge-connectivity augmentation problem: find a smallest set of directed edges whose addition to the given directed graph makes it strongly $k$-connected. We improve the deterministic complexity for this problem from $O(k δ(m+δn)\log n)$ [Gabow, STOC 1994] to $O(k δm \log n)$. A similar approach with the same complexity also works for the undirected version of the problem.

cs.DS↗

arXiv2025

Bounded indegree $k$-forests problem and a faster algorithm for directed graph augmentation

We consider two problems for a directed graph $G$, which we show to be closely related. The first one is to find $k$ edge-disjoint forests in $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We describe a min-max characterization for this problem and show that it can be solved in $O(k δm \log n)$ time, where $(n,m)$ is the size of $G$ and $δ$ is the difference between $k$ and the edge connectivity of the graph. The second problem is the directed edge-connectivity augmentation problem, which has been extensively studied before: find a smallest set of directed edges whose addition to the graph makes it strongly $k$-connected. We improve the complexity for this problem from $O(k δ(m+δn)\log n)$ [Gabow, STOC 1994] to $O(k δm \log n)$, by exploiting our solution for the first problem. A similar approach with the same complexity also works for the undirected version of the problem.

cs.DS↗

arXiv2024

Majority Dynamics and Internal Partitions of Random Regular Graphs: Experimental Results

This paper focuses on Majority Dynamics in sparse graphs, in particular, as a tool to study internal cuts. It is known that, in Majority Dynamics on a finite graph, each vertex eventually either comes to a fixed state, or oscillates with period two. The empirical evidence acquired by simulations suggests that for random odd-regular graphs, approximately half of the vertices end up oscillating with high probability. We notice a local symmetry between oscillating and non-oscillating vertices, that potentially can explain why the fraction of the oscillating vertices is concentrated around $\frac{1}{2}$. In our simulations, we observe that the parts of random odd-regular graph under Majority Dynamics with high probability do not contain $\lceil \frac{d}{2} \rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to prove that internal cuts exist in odd-regular graphs almost surely. However, we suggest a modification of Majority Dynamics, that yields parts with desired cores with high probability.

math.CO↗

arXiv2022

Synchronization on star-like graphs and emerging $\mathbb{Z}_{p}$ symmetries at strong coupling

We discuss the aspects of synchronization on inhomogeneous star-like graphs with long rays in Kuramoto model framework. We assume the positive correlation between internal frequencies and degrees for all nodes which supports the abrupt first order synchronization phase transition. It is found that different ingredients of the graph get synchronized at different critical couplings. Combining numerical and analytic tools we evaluate all critical couplings for the long star graph. Surprisingly it is found that at strong coupling there are discrete values of coupling constant which support the synchronized states with emerging $\mathbb{Z}_{p}$ symmetries. The stability of synchronized phase is discussed and the interpretation of phase with emerging $\mathbb{Z}_{p}$ symmetry for the Josephson array on long star graph is mentioned.

cond-mat.dis-nn↗