Minimum Label $k$-Cut on Planar Graphs: Complexity and Color Structure
Yuanhao Wang, Wei Wang
AI 速览：研究底层图结构与颜色类结构如何影响最小标签 k-割问题的参数化复杂性与可解性边界。
Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.
Understanding the internal routing of visual concepts is crucial for the safe and controllable adaptation of generative models. While concept localization has been widely studied in diffusion models, the emerging paradigm of next-scale visual autoregressive models remains largely unexplored. In this paper, we introduce LoCo, the first model-agnostic framework to localize where and when specific concepts emerge within autoregressive models. Driven by the native coarse-to-fine nature of next-scale generation, our method precisely maps conceptual knowledge across three distinct dimensions: Layer, Scale, and Position. To systematically localize and evaluate concept routing without context bias, we propose LoCoBench, a comprehensive dataset spanning 10 diverse categories. Extensive probing on image and video autoregressive models like Infinity, HunyuanImage-3.0, and InfinityStar shows that the localized positions are both interpretable and causally related to concept emergence. Building on these insights, we apply our localization method to three key applications: concept erasure, model personalization, and adversarial concept injection. Experiments demonstrate that our targeted intervention achieves state-of-the-art performance, substantially reducing computational overhead while preserving benign utility. Overall, our findings offer insights into how conceptual knowledge is routed during autoregressive generation, introducing a practical pathway for more interpretable, efficient, and secure adaptation.