Organizations: Department of Mathematics, University of California, Irvine
Abstract
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.
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.
Multi-agent combinatorial optimization problems are notoriously challenging due to their NP-hard nature. Recent parallel autoregressive neural solvers improve inference efficiency by allowing agents to make decisions simultaneously, but their performance often degrades on large-scale instances. This is largely attributable to weak modeling of local geometric structures and the fact that conflicting task selections are handled only after action generation. To address these limitations, we propose GeoPAR, a geometry-guided parallel autoregressive reinforcement learning framework for scalable multi-agent combinatorial optimization. GeoPAR integrates three key components: (1) a projection-window sparse geometry mechanism that builds lightweight local candidate neighborhoods through multi-directional projections, (2) sparse edge-biased attention that injects these geometric relations into node representations, and (3) cache-guided conflict-aware assignment that reuses the geometric cache during decoding to suppress duplicate selections of exclusive tasks. Experiments on heterogeneous vehicle routing and open multi-depot pickup-and-delivery problems show that GeoPAR improves large-scale zero-shot generalization while substantially reducing rollout steps and maintaining efficient inference.