**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.

\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…

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $...

**Authors:** Amatya Sharma, Santhoshini Velusamy Non-redundancy, introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020), is a structural parameter for Constraint Satisfaction Problems ($\mathsf{CSPs}$) that governs kernelization, exact and approximate sparsification, and exact streaming complexity. It is the largest size of a $\mathsf{CSP}$ instance admitting no smaller subinstance with the same satisfying assignments. We study non-redundancy $\mathsf{NRD}_n(R)$ for Boolean symmetric $\mathsf{CSPs}$ defined by an $r$-ary relation $R$ whose value depends only on Hamming weight. An instance of $\mathsf{CSP}(R)$ has $n$ variables and constraints given by $r$-tuples; a constraint is satisfied exactly when the induced tuple lies in $R$. This class includes natural predicates such as cuts and $k$-SAT clauses. Our main result is a near-complete classification of the asymptotic growth of $\mathsf{NRD}_n(R)$ for symmetric Boolean predicates of arity at most $5$. Using computational experiments and algebraic upper- and lower-bound criteria, we resolve every predicate of arity at most $4$ and all but two predicates of arity $5$. For upper bounds, we introduce $t$-balancedness, a lifted, higher-degree version of the balancedness notion of Chen, Jansen, and Pieterse (Algorithmica 2020). We prove that $t$-balancedness is equivalent to the existence of degree-$t$ multilinear polynomials capturing $R$, and hence implies $\mathsf{NRD}_n(R)=O(n^t)$. For lower bounds, we use Carbonnel's (CP 2022) framework: predicates admitting a special reduction from $k$-ary OR inherit OR's lower bound $Ω(n^k)$. The only unresolved arity-$5$ predicates in our framework have bounds $Ω(n^2)$ and $O(n^3)$; we reduce their exact classification to natural extremal set-system questions.

**Authors:** Ishay Haviv We study the kernel complexity of constraint satisfaction problems over a finite domain, parameterized by the number of variables, whose constraint language consists of two relations: the non-equality relation and an additional permutation-invariant relation $R$. We establish a conditional lower bound on the kernel size in terms of the largest arity of an OR relation definable from $R$. Building on this, we investigate the kernel complexity of uniformly rainbow free coloring problems. In these problems, for fixed positive integers $d$, $\ell$, and $q \geq d$, we are given a graph $G$ on $n$ vertices and a collection $\cal F$ of $\ell$-tuples of $d$-subsets of its vertex set, and the goal is to decide whether there exists a proper coloring of $G$ with $q$ colors such that no $\ell$-tuple in $\cal F$ is uniformly rainbow, that is, no tuple has all its sets colored with the same $d$ distinct colors. We determine, for all admissible values of $d$, $\ell$, and $q$, the infimum over all values $η$ for which the problem admits a kernel of size $O(n^η)$, under the assumption $\mathsf{NP} \nsubseteq \mathsf{coNP/poly}$. As applications, we obtain nearly tight bounds on the kernel complexity of various coloring problems under diverse settings and parameterizations. This includes graph coloring problems parameterized by the vertex-deletion distance to a disjoint union of cliques, resolving a question of Schalken (2020), as well as uniform hypergraph coloring problems parameterized by the number of vertices, extending results of Jansen and Pieterse (2019) and Beukers (2021).

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $...

**Authors:** Niels Holtgrefe, Jannik Schestag, Norbert Zeh In the Maximize Phylogenetic Diversity problem, we are given a phylogenetic tree that represents the genetic proximity of species, and we are asked to select a subset of species of maximum phylogenetic diversity to be preserved through conservation efforts, subject to budgetary constraints that allow only k species to be saved. This neglects that it is futile to preserve a predatory species if we do not also preserve at least a subset of the prey it feeds on. Thus, in the Optimizing PD with Dependencies ($ε$-PDD) problem, we are additionally given a food web that represents the predator-prey relationships between species. The goal is to save a set of k species of maximum phylogenetic diversity such that for every saved species, at least one of its prey is also saved. This problem is NP-hard even when the phylogenetic tree is a star. The $α$-PDD problem alters PDD by requiring that at least some fraction $α$ of the prey of every saved species are also saved. In this paper, we study the parameterized complexity of $α$-PDD. We prove that the problem is W[1]-hard and in XP when parameterized by the solution size k, the diversity threshold D, or their complements. When parameterized by the vertex cover number of the food web, $α$-PDD is fixed-parameter tractable (FPT). A key measure of the computational difficulty of a problem that is FPT is the size of the smallest kernel that can be obtained. We prove that, when parameterized by the distance to clique, 1-PDD admits a linear kernel. Our main contribution is to prove that $α$-PDD does not admit a polynomial kernel when parameterized by the vertex cover number plus the diversity threshold D, even if the phylogenetic tree is a star. This implies the non-existence of a polynomial kernel for $α$-PDD also when parameterized by a range of structural parameters of the food web, such as its dist[...]

**Abstract:** This paper introduces a novel framework, Hyperdimensional Temporal Graph Kernelization for Stochastic Differential Equation Approximation (HTGK-SDE), for significantly reducing computational complexity and improving accuracy in solving high-dimensional stochastic differential equations (SDEs). Drawing from advancements in combinatorial optimization, hyperdimensional computing, and temporal graph learning, we leverage the power of graph kernels to represent SDE solutions as hypervectors navigable within a vast, high-dimensional space.

**Abstract:** This paper presents a novel framework, Scalable Deep Kernelization (SDK), for improving the accuracy and robustness of Emotion State Recognition (ESR) systems based on electroencephalography (EEG) data, while addressing ethical concerns surrounding data privacy and algorithmic bias. SDK leverages a combination of deep learning and kernel methods to extract both physiological and semantic features from EEG signals, facilitating accurate emotion classification across diverse demographic groups and minimizing susceptibility to noise and artifacts.