Semi-supervised learning (SSL) on graphs is critical in applications where labeled data are scarce and costly, yet existing graph-based methods often degrade under extreme label sparsity or class imbalance, yielding trivial or unstable solutions. We introduce \textbf{CombCut}, the first exact combinatorial optimization framework for multi-class graph-based semi-supervised learning that operates directly on binary one-hot assignments, without any convex relaxation or heuristic volume constraints. By employing a minorization–maximization (MM) scheme, CombCut transforms each step into a structured linear assignment problem solved efficiently via network-flow algorithms. Total unimodularity guarantees integral iterates, and our theoretical analysis establishes both monotonic ascent of the true discrete objective and convergence of every limit point to a Karush–Kuhn–Tucker (KKT) stationary solution of the original combinatorial problem. Our approach requires no hyperparameter tuning and scales near-linearly in the number of vertices. Empirical evaluation on MNIST, Fashion-MNIST, and CIFAR-10 with as few as 1–5 labels per class shows that CombCut excels in worst-case labeling scenarios, significantly outperforming state-of-the-art graph-SSL baselines and yielding more stable and accurate label propagation under severe supervision constraints.

While significant progress has been made in designing algorithms that minimize regret in online decision-making, real-world scenarios often introduce additional complexities, with missing outcomes perhaps among the most challenging ones. Overlooking this aspect or simply assuming random missingness invariably leads to biased estimates of the rewards and may result in linear regret. Despite the practical relevance of this challenge, no rigorous methodology currently exists for systematically handling missingness, especially when the missingness mechanism is not random. In this paper, we address this gap in the context of multi-armed bandits (MAB) with missing outcomes by analyzing the impact of different missingness mechanisms on achievable regret bounds. We introduce algorithms that account for missingness under both missing at random (MAR) and missing not at random (MNAR) models. Through both analytical and simulation studies, we demonstrate the drastic improvements in decision-making by accounting for missingness in these settings.