We study certain extremal problems in combinatorial geometry that ask about configurations of points in an n×n grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from O(n3) to O(n2). To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly 1.8n for grids of size 82≤n≤119. For the Smallest Complete Set problem, we find configurations of size roughly 0.95n, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.
We study the Compositional Geometry Routing Problem (CGRP), a unified superclass of traditional routing problems that covers point-only, line-only, area-only, and arbitrary hybrid task geometries, providing a broad abstraction for real-world routing scenarios. Beyond standard point-based routing, CGRP with non-point tasks can be inherently asymmetric, tightly coupled travel routes with the intrinsic path, and enlarges the action space with numerous feasible yet often irrelevant options, thereby posing significant challenges for both representation learning and decision-making. To address these challenges, we propose DiCon, a differential attention-assisted solver with contrastive learning, as a plug-and-play framework that tackles the problem from two complementary angles. First, we introduce a differential attention mechanism that actively suppresses the probability mass on less competitive candidate actions. Second, we design a double-level contrastive learning objective to promote robust global instance representations and regularize geometry-aware task representations. Extensive experiments demonstrate that DiCon achieves strong performance, broad versatility, and superior generalization across diverse CGRP instances with different compositions.
Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a defining challenge in discrete geometry. As the local analogue of Hilbert's 18th problem, it has profound implications across geometry, number theory and information theory. Although lattices and codes have achieved significant progress, the field is confined to isolated extremal configurations, leaving underlying geometric principles obscured. Here we shift the object to the broader extremal configuration space, thereby opening a new path for the Kissing Number Problem. Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable. Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products. These findings reveal the first explicit spherical-code realization of the Fischer group Fi22, extend the classical Euclidean representation of subgroup structure, and directly inspire subsequent breakthroughs by mathematicians. Overall, the work provides an early example of AI-driven progress on a Hilbert-calibre problem, showing how reinforcement learning advances mathematical discovery by unlocking more expressive objects.
Traditional heuristic solvers for the 2D irregular nesting problem share a fundamental limitation: they are blind to polygon geometry, relying on guided brute-force to navigate the continuous placement space with minimal geometrical guidance. In this paper, we argue that Reinforcement Learning is uniquely positioned to overcome this bottleneck. By pairing an optimization policy with a geometry-aware neural encoder, an agent can automatically discover rich geometric priors directly from data, utilizing these learned intuitions to strategically guide exploration. To realize this, we introduce the Polygons Transformer (PoT), a novel architecture that encodes 2D continuous vector geometries while allowing cross-polygons attention. We couple this novel architecture with a Combinatorial Optimization Reinforcement Learning (CORL) training framework to find optimal solutions. To support this paradigm, we release an open-source training dataset derived from complex geographic contours alongside a dedicated evaluation benchmark. Our empirical validation demonstrates that our trained agent achieves area utilization performance highly competitive with Sparrow, the state-of-the-art heuristic solver, proving that reinforcement learning can successfully discover and exploit geometric awareness for precise spatial tasks.