Hybrid Mesh-Gaussian rendering combines compact textured surfaces with flexible 3D Gaussian primitives, offering an efficient alternative to purely Gaussian scene representations for structured environments. Existing scene-reconstruction hybrids commonly follow an additive formulation, where Gaussian residuals compensate for errors left by the mesh. However, not every Mesh-Gaussian conflict is caused by missing content: residual geometry errors, occlusion mismatch, texture-projection artifacts, and view-dependent appearance can make the same mesh content consistent in one view but inconsistent in another. We therefore formulate hybrid rendering as a multi-view inconsistency problem and present Gaussolve, a subtract-and-refill framework that selectively dissolves inconsistent mesh content before residual Gaussians refill the revealed regions. We introduce a lightweight Neural Dissolve Field, parameterized by scalar spherical-harmonic Gaussians, to learn where and from which viewing directions mesh content should be dissolved. A continuous transmittance fusion rule progressively exposes residual Gaussians behind the dissolved mesh, preserving their reconstruction gradients and avoiding hard visibility discontinuities. Experiments on structured and semi-structured scenes demonstrate higher visual fidelity with fewer Gaussian primitives and faster rendering. Project page: https://ooosn.github.io/gaussians-can-subtract/
We study exact exponential-time algorithms for 3- and 4-coloring. For 3-coloring, the best previously published running-time bound is $O^*(1.3289^n)$, due to Eppstein (SODA 2001). We improve this bound to $O^*(1.3204^n)$ deterministically and $O^*(1.3159^n)$ using randomization. For 4-coloring, the best previously known running-time bound is $O^*(1.7159^n)$, due to Wu, Gu, Jiang, Shao, and Xu (ESA 2024). We improve this bound to $O^*(1.6975^n)$ deterministically and $O^*(1.6933^n)$ using randomization. Our algorithms for both problems combine constraint satisfaction algorithms with dynamic programming over tree and path decompositions, exploiting their complementary strengths on different graph structures. For 6-coloring, the best published randomized running-time bound is exponentially smaller than the best published deterministic bound. For 3- and 4-coloring, however, no such improvement through randomization was previously known. To the best of our knowledge, our results provide the first randomized running-time bounds for these two problems that exponentially improve upon the best previously published deterministic bounds. This raises the question of whether randomization offers an inherent exponential advantage for 3- and 4-coloring.