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Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs
Pan Peng, Yuyang Wang
https://arxiv.org/abs/2602.07394 https://arxiv.org/pdf/2602.07394 https://arxiv.org/html/2602.07394
arXiv:2602.07394v1 Announce Type: new
Abstract: We study \emph{local computation algorithms (LCAs)} for constructing spanning trees. In this setting, the goal is to locally determine, for each edge $ e \in E $, whether it belongs to a spanning tree $ T $ of the input graph $ G $, where $ T $ is defined implicitly by $ G $ and the randomness of the algorithm. It is known that LCAs for spanning trees do not exist in general graphs, even for simple graph families. We identify a natural and well-studied class of graphs -- \emph{expander graphs} -- that do admit \emph{sublinear-time} LCAs for spanning trees. This is perhaps surprising, as previous work on expanders only succeeded in designing LCAs for \emph{sparse spanning subgraphs}, rather than full spanning trees. We design an LCA with probe complexity $ O\left(\sqrt{n}\left(\frac{\log^2 n}{\phi^2} d\right)\right)$ for graphs with conductance at least $ \phi $ and maximum degree at most $ d $ (not necessarily constant), which is nearly optimal when $\phi$ and $d$ are constants, since $\Omega(\sqrt{n})$ probes are necessary even for expanders. Next, we show that for the natural class of \emph{\ER graphs} $ G(n, p) $ with $ np = n^{\delta} $ for any constant $ \delta > 0 $ (which are expanders with high probability), the $ \sqrt{n} $ lower bound can be bypassed. Specifically, we give an \emph{average-case} LCA for such graphs with probe complexity $ \tilde{O}(\sqrt{n^{1 - \delta}})$.
Finally, we extend our techniques to design LCAs for the \emph{minimum spanning tree (MST)} problem on weighted expander graphs. Specifically, given a $d$-regular unweighted graph $\bar{G}$ with sufficiently strong expansion, we consider the weighted graph $G$ obtained by assigning to each edge an independent and uniform random weight from $\{1,\ldots,W\}$, where $W = O(d)$. We show that there exists an LCA that is consistent with an exact MST of $G$, with probe complexity $\tilde{O}(\sqrt{n}d^2)$.
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#MehganHeaneyGrier #DavidChokachi
Season 9 Episode 12 "The Big Blue"
#Randombaywatch

Digitalisierung klingt einfach. Ist sie aber nicht.
In unserem Podcast #9vor9 diskutieren wir mit Holger Klötzner, Stadtrat für Digitalisierung, und Dennis Schwalbach aus dem IT-Maschinenraum der Stadt Darmstadt über Formulare, Fachverfahren – und die Umlaufmappe 📁.
Was bedeutet digitale Verwaltung wirklich?
Wo hakt es noch?
Reinhören unter

"On Tuesday, the Department of Education announced that it had reached a settlement with the state of Missouri to end the Saving on a Valuable Education (SAVE) program, which allowed more than 7 million mostly low-income Americans to reduce their federal student loan payments."
Trump admin blasted for 'unrelenting push to jack up costs' with new 'backroom deal' - Raw Story
https://www.rawstory.com/linda-mcmahon-2674384567/

«LLMs are cliché machines, trained on a resilient human weakness for generating maximum content with minimum effort.»
Bingo.
Unfortunately, this too hits the nail on the head: «Bad art is something human beings love to do, in vast numbers. It’s part of who we are, and when abandoned by inspiration we trust in the same methods we’ve programmed into LLMs.»

Sure, AI can ‘do’ writing. But memoir? Not so much | Aeon Essays
As AI’s endless clichés continue to encroach on human art, the true uniqueness of our creativity is becoming ever clearer

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Neighborhood-Aware Graph Labeling Problem
Mohammad Shahverdikondori, Sepehr Elahi, Patrick Thiran, Negar Kiyavash
https://arxiv.org/abs/2602.08098 https://arxiv.org/pdf/2602.08098 https://arxiv.org/html/2602.08098
arXiv:2602.08098v1 Announce Type: new
Abstract: Motivated by optimization oracles in bandits with network interference, we study the Neighborhood-Aware Graph Labeling (NAGL) problem. Given a graph $G = (V,E)$, a label set of size $L$, and local reward functions $f_v$ accessed via evaluation oracles, the objective is to assign labels to maximize $\sum_{v \in V} f_v(x_{N[v]})$, where each term depends on the closed neighborhood of $v$. Two vertices co-occur in some neighborhood term exactly when their distance in $G$ is at most $2$, so the dependency graph is the squared graph $G^2$ and $\mathrm{tw}(G^2)$ governs exact algorithms and matching fine-grained lower bounds. Accordingly, we show that this dependence is inherent: NAGL is NP-hard even on star graphs with binary labels and, assuming SETH, admits no $(L-\varepsilon)^{\mathrm{tw}(G^2)}\cdot n^{O(1)}$-time algorithm for any $\varepsilon>0$. We match this with an exact dynamic program on a tree decomposition of $G^2$ running in $O\!\left(n\cdot \mathrm{tw}(G^2)\cdot L^{\mathrm{tw}(G^2) 1}\right)$ time. For approximation, unless $\mathsf{P}=\mathsf{NP}$, for every $\varepsilon>0$ there is no polynomial-time $n^{1-\varepsilon}$-approximation on general graphs even under the promise $\mathrm{OPT}>0$; without the promise $\mathrm{OPT}>0$, no finite multiplicative approximation ratio is possible. In the nonnegative-reward regime, we give polynomial-time approximation algorithms for NAGL in two settings: (i) given a proper $q$-coloring of $G^2$, we obtain a $1/q$-approximation; and (ii) on planar graphs of bounded maximum degree, we develop a Baker-type polynomial-time approximation scheme (PTAS), which becomes an efficient PTAS (EPTAS) when $L$ is constant.
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