Exclusive | The Sudden Fall of OpenAI’s Most Hyped Product Since ChatGPT - WSJ
Exclusive | The Sudden Fall of OpenAI’s Most Hyped Product Since ChatGPT WSJ
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Round-efficient Fully-scalable MPC algorithms for k-Means
arXiv:2604.00954v1 Announce Type: new Abstract: We study Euclidean $k$-Means under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully-scalable} setting. Our main result is a fully-scalable $O((\log n/\log\log n)^2)$-approximation in $O(1)$ rounds. Previously, fully-scalable algorithms for $k$-Means either run in super-constant $O(\log\log n \cdot \log\log\log n)$ rounds, albeit with a better $O(1)$-approximation [Cohen-Addad et al., SODA'26], or suffer from bicriteria guarantees [Bhaskara and Wijewardena, ICML'18; Czumaj et al., ICALP'24]. Our algorithm also gives an $O(\log n/\log\log n)$-approximation for $k$-Median, which improves a recent $O(\log n)$-approximation [Goranci et al., SODA'26], and this $o(\log n)$ ratio breaks the fundamental barrier of tree embed
Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support
arXiv:2604.00219v1 Announce Type: new Abstract: Given an unweighted graph $G$, the *minimum $r$-dominating set problem* asks for the smallest-cardinality subset $S$ such that every vertex in $G$ is within radius $r$ of some vertex in $S$. While the $r$-dominating set problem on planar graphs admits a PTAS from Baker's shifting/layering technique when $r$ is constant, it becomes significantly harder when $r$ can depend on $n$. Under the Exponential-Time Hypothesis, Fox-Epstein et al. [SODA 2019] showed that no efficient PTAS exists for the unbounded $r$-dominating set problem on planar graphs. One may also consider the harder *vertex-weighted metric $r$-dominating set*, where edges have lengths, vertices have positive weights, and the goal is to find an $r$-dominating set of minimum total w
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