Combinatorial Optimization (CO) addresses many important problems, including the challenging Maximum Independent Set (MIS) problem. Alongside exact and heuristic solvers, differentiable approaches have emerged, often using continuous relaxations of ReLU-based or quadratic objectives. Noting that an MIS in a graph is a Maximum Clique (MC) in its complement, we propose a new quadratic formulation for MIS by incorporating an MC term, improving convergence and exploration. We show that every maximal independent set corresponds to a local minimizer, derive conditions for the MIS size, and characterize stationary points. To solve our non-convex objective, we propose solving parallel multiple initializations using momentum-based gradient descent, complemented by an efficient MIS checking criterion derived from our theory. Therefore, we dub our method as parallelized Clique-Informed Quadratic Optimization for MIS (pCQO-MIS). Our experimental results demonstrate the effectiveness of the proposed method compared to exact, heuristic, sampling, and data-centric approaches. Notably, our method avoids the out-of-distribution tuning and reliance on (un)labeled data required by data-centric methods, while achieving superior MIS sizes and competitive runtime relative to their inference time. Additionally, a key advantage of pCQO-MIS is that, unlike exact and heuristic solvers, the runtime scales only with the number of nodes in the graph, not the number of edges.

翻译：组合优化（CO）处理包括具有挑战性的最大独立集（MIS）问题在内的许多重要问题。除了精确求解器和启发式求解器之外，可微分方法已经出现，通常使用基于ReLU或二次目标的连续松弛。注意到图中的一个MIS是其补图中的最大团（MC），我们通过引入一个MC项提出了一种新的MIS二次规划形式，从而改善了收敛性和探索能力。我们证明了每个极大独立集都对应一个局部极小值点，推导了MIS规模的充要条件，并刻画了驻点特性。为求解我们的非凸目标函数，我们提出使用基于动量的梯度下降法并行求解多个初始化点，并辅以从理论推导出的高效MIS检验准则。因此，我们将该方法命名为并行化团信息二次优化求解MIS（pCQO-MIS）。实验结果表明，与精确方法、启发式方法、采样方法以及以数据为中心的方法相比，所提方法具有显著优势。值得注意的是，我们的方法避免了以数据为中心方法所需的分布外调优和对（未）标记数据的依赖，同时在MIS规模上表现更优，且推理时间具有竞争力。此外，pCQO-MIS的一个关键优势在于，与精确求解器和启发式求解器不同，其运行时间仅随图中节点数量而非边数量增长。