Neural Information Processing Systems http://nips.cc/

Addressing irregular cutting and packing (C&P) optimization problems poses two distinct challenges: the geometric challenge of determining whether or not an item can be placed feasibly at a certain position, and the optimization challenge of finding a good solution according to some objective function. Until now, those tackling such problems have had to address both challenges simultaneously, requiring two distinct sets of expertise and a lot of research & development effort. One way to lower this barrier is to decouple the two challenges. In this paper we introduce a powerful collision detection engine (CDE) for 2D irregular C&P problems which assumes full responsibility for the geometric challenge. The CDE (i) allows users to focus with full confidence on their optimization challenge by abstracting geometry away and (ii) enables independent advances to propagate to all optimization algorithms built atop it. We present a set of core principles and design philosophies to model a general and adaptable CDE focused on maximizing performance, accuracy and robustness. These principles are accompanied by a concrete open-source implementation called jagua-rs. This paper together with its implementation serves as a catalyst for future advances in irregular C&P problems by providing a solid foundation which can either be used as it currently exists or be further improved upon. Funding: This research was supported by the Research Foundation -- Flanders (FWO) under grant number 1S71222N and K804824N.

Proof of Theorem 2. The Kantorovich Rubinstein (KR) distance is defined as follows. The optimal partial transport distance is defined as follows. We prove the contraposition of Theorem 2 for the KR distance. We solve the BHCP problem using this algorithm.

We thank all reviewers for their positive feedback. Figalli's distance, our algorithm computes it efficiently by setting λ We compare the distributions of crime locations. Our method is the first UOT method that can handle million-scale datasets. The cost is the ground distance between the centers of regions of the quadtree. GKR is reduced to the standard OT, thus metric.

Accurate agricultural weed mapping using UAVs is crucial for precision farming applications. Traditional methods rely on orthomosaic stitching from rigid flight paths, which is computationally intensive and time-consuming. Gaussian Process (GP)-based mapping offers continuous modelling of the underlying variable (i.e. weed distribution) but requires discretisation for practical tasks like path planning or visualisation. Current implementations often default to quadtrees or gridmaps without systematically evaluating alternatives. This study compares five discretisation methods: quadtrees, wedgelets, top-down binary space partition (BSP) trees using least square error (LSE), bottom-up BSP trees using graph merging, and variable-resolution hexagonal grids. Evaluations on real-world weed distributions measure visual similarity, mean squared error (MSE), and computational efficiency. Results show quadtrees perform best overall, but alternatives excel in specific scenarios: hexagons or BSP LSE suit fields with large, dominant weed patches, while quadtrees are optimal for dispersed small-scale distributions. These findings highlight the need to tailor discretisation approaches to weed distribution patterns (patch size, density, coverage) rather than relying on default methods. By choosing representations based on the underlying distribution, we can improve mapping accuracy and efficiency for precision agriculture applications.