🤖 AI Summary
本文通过使用连接树表示法，开发了一种精确线性规划公式来解决有界深度分类树问题，并提出了两种解决方案：列生成和消息传递。
📝 Abstract
We develop an exact linear programming (LP) formulation for bounded-depth classification trees with binary features, using a junction-tree representation. The formulation is integral and supports recursive subtree optimization. Exact reductions make the model smaller while preserving the optimal value and recovery of an optimal tree. The reduced model supports two solution methods: column generation and message passing. Column generation solves integral restricted LPs and uses bounds over the full feasible domain to certify optimality. Message passing recursively combines optimal subtree costs. Both methods solve common subtree problems that, once the preceding tree decisions are fixed, can be evaluated independently and in parallel. Computational experiments show that the exact reductions substantially reduce the size of the junction-tree formulation. The resulting linear programming formulation certifies instances for which the tested mixed-integer formulation does not establish optimality within the same computational budget, while the column-generation and message-passing methods certify more instances and achieve an order-of-magnitude reduction in geometric-mean runtime relative to an existing state-of-the-art exact method for optimal classification trees.