Combinatorial Optimization (CO) plays a crucial role in addressing various significant problems, among them the challenging Maximum Independent Set (MIS) problem. In light of recent advancements in deep learning methods, efforts have been directed towards leveraging data-driven learning approaches, typically rooted in supervised learning and reinforcement learning, to tackle the NP-hard MIS problem. However, these approaches rely on labeled datasets, exhibit weak generalization, and often depend on problem-specific heuristics. Recently, ReLU-based dataless neural networks were introduced to address combinatorial optimization problems. This paper introduces a novel dataless quadratic neural network formulation, featuring a continuous quadratic relaxation for the MIS problem. Notably, our method eliminates the need for training data by treating the given MIS instance as a trainable entity. More specifically, the graph structure and constraints of the MIS instance are used to define the structure and parameters of the neural network such that training it on a fixed input provides a solution to the problem, thereby setting it apart from traditional supervised or reinforcement learning approaches. By employing a gradient-based optimization algorithm like ADAM and leveraging an efficient off-the-shelf GPU parallel implementation, our straightforward yet effective approach demonstrates competitive or superior performance compared to state-of-the-art learning-based methods. Another significant advantage of our approach is that, unlike exact and heuristic solvers, the running time of our method scales only with the number of nodes in the graph, not the number of edges.

翻译：组合优化在解决各类重要问题中扮演着关键角色，其中极具挑战性的最大独立集问题便是典型代表。随着深度学习方法的最新进展，学界开始尝试利用数据驱动的学习方法（通常基于监督学习与强化学习）来应对这一NP难问题。然而，这些方法依赖标注数据集，泛化能力较弱，且往往需要借助问题特定的启发式策略。近期，基于ReLU的无数据神经网络被提出用于求解组合优化问题。本文提出一种新颖的无数据二次神经网络架构，针对最大独立集问题设计了连续二次松弛形式。值得注意的是，我们的方法通过将给定的MIS实例本身作为可训练对象，完全消除了对训练数据的需求。具体而言，我们利用MIS实例的图结构与约束条件来定义神经网络的结构与参数，使得在固定输入上训练该网络即可获得问题解，这使其与传统监督学习或强化学习方法形成显著区别。通过采用基于梯度的优化算法（如ADAM）并利用高效的现成GPU并行实现，我们这种简洁而有效的方法相较于最先进的基于学习方法展现出具有竞争力乃至更优的性能。本方法的另一重要优势在于：与精确求解器和启发式求解器不同，我们的方法运行时间仅随图中节点数量增长，而不受边数影响。