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cs.DS updates on arXiv.org

PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Triangle Detection in Worst-Case Sparse Graphs via Local ...
Hongyi Duan, Jian'an Zhang · 2025-09-03 · via cs.DS updates on arXiv.org

We present a non-algebraic, locality-preserving framework for triangle detection in worst-case sparse graphs. Our algorithm processes the graph in $O(\log n)$ independent layers and partitions incident edges into prefix-based classes where each class maintains a 1-sparse triple over a prime field. Potential witnesses are surfaced by pair-key (PK) alignment, and every candidate is verified by a three-stage, zero-false-positive pipeline: a class-level 1-sparse consistency check, two slot-level decodings, and a final adjacency confirmation. \textbf{To obtain single-run high-probability coverage, we further instantiate $R=c_G\log n$ independent PK groups per class (each probing a constant number of complementary buckets), which amplifies the per-layer hit rate from $Θ(1/\log n)$ to $1-n^{-Ω(1)}$ without changing the accounting.} A one-shot pairing discipline and class-term triggering yield a per-(layer,level) accounting bound of $O(m)$, while keep-coin concentration ensures that each vertex retains only $O(d^+(x))$ keys with high probability. Consequently, the total running time is $O(m\log^2 n)$ and the peak space is $O(m\log n)$, both with high probability. The algorithm emits a succinct Seeds+Logs artifact that enables a third party to replay all necessary checks and certify a NO-instance in $\tilde O(m\log n)$ time. We also prove a $Θ(1/\log n)$ hit-rate lower bound for any single PK family under a constant-probe local model (via Yao)--motivating the use of $Θ(\log n)$ independent groups--and discuss why global algebraic convolutions would break near-linear accounting or run into fine-grained barriers. We outline measured paths toward Las Vegas $O(m\log n)$ and deterministic near-linear variants.