**Authors:** Shubhada Aute, Fahad Panolan, Geevarghese Philip An edge weighting of a graph induces a coloring of its vertices in which the color of a vertex is the total weight of the edges incident with it. Such an edge weighting is proper if adjacent vertices always receive distinct colors. Deciding whether a graph admits a proper weighting is known to be NP-complete for the weight set $\\{0,1\\}$, and also for $\\{1,2\\}$. In recent work (arXiv:2604.12363) we showed that both problems are FPT parameterized by the vertex cover number $k$, but it was open -- to the best of our knowledge -- whether either parameterized problem had a polynomial kernel. In this work, we show that both problems have polynomial kernels when parameterized by $k$. We also show that both problems are W[1]-hard parameterized by treedepth, answering another question from our earlier work. We then study the pre-weighted versions of the two problems, in which the weights of some edges are fixed in advance, and the task is to extend the assignment to a proper weighting of the whole graph. We show that both pre-weighted problems are FPT parameterized by the vertex cover number $k$. For the $\\{1,2\\}$ version the running time is $2^{O(k \log k)} \cdot n$; for the $\\{0,1\\}$ version we obtain the same running time when every pre-weight is $1$, and a slower FPT algorithm in the general case. We also show that both pre-weighted problems are W[1]-hard parameterized by either of (i) the feedback vertex set number or (ii) the treedepth of the input graph. Since a graph with no pre-assigned weights is a special case, our algorithms for the pre-weighted versions solve the two original problems as well, in time $2^{O(k \log k)} \cdot n$, significantly improving on the bound of $2^{O(k^4)} \cdot n^{O(1)}$ from our earlier work.

**Originality:** This research proposes a novel framework that dynamically adapts the metric space used to represent the state space of ergodic dynamical systems by leveraging machine learning techniques. Unlike traditional approaches relying on fixed metrics or handcrafted features, our method learns an optimal metric that maximizes predictive accuracy, particularly for chaotic, high-dimensional systems. This methodology integrates adaptive kernel methods with rigorous statistical validation, substantially improving the forecasting precision of complex chaotic behavior.

The IBS discrete mathematics group welcomes Dr. Roohani Sharma, a new research fellow at the IBS Discrete Mathematics Group from May 1, 2025. She received her Ph.D. from the Institute of Mathematical Sciences, Chennai, India, under the supervision of Prof. Saket Saurabh. She is interested in parameterized complexity and kernelization. Previously, she was a researcher at the University of Bergen, Norway, and a Lise-Meitner Post-doctoral Fellow at the Max Planck Institute for Informatics in Germany.

\textsc{$H$-Packing} asks whether a graph $G$ contains $k$ vertex-disjoint copies of a fixed pattern graph $H$. Via the standard reduction to \textsc{$d$-Set Packing}, one obtains generic kernels with $O(k^{|V(H)|-1})$ vertices and $O(k^{|V(H)|})$ edges. We revisit the question of beating these bou…