The length-constrained cycle partition problem (LCCP) is a graph optimization problem in which a set of nodes must be partitioned into a minimum number of cycles. Every node is associated with a critical time and the length of every cycle must not exceed the critical time of any node in the cycle. We formulate LCCP as a set partitioning model and solve it using an exact branch-and-price approach. We use a dynamic programming-based pricing algorithm to generate improving cycles, exploiting the particular structure of the pricing problem for efficient bidirectional search and symmetry breaking. Computational results show that the LP relaxation of the set partitioning model produces strong dual bounds and our branch-and-price method improves significantly over the state of the art. It is able to solve closed instances in a fraction of the previously needed time and closes 13 previously unsolved instances, one of which has 76 nodes, a notable improvement over the previous limit of 52 nodes.

翻译：带长度约束的环分割问题（LCCP）是一类图优化问题，要求将节点集划分为最少数量的环。每个节点关联一个临界时间，且每个环的长度不得超过环内任意节点的临界时间。本文将LCCP建模为集合分割模型，并采用精确的分支定价方法求解。我们设计了一种基于动态规划的定价算法以生成改进环，通过利用定价问题的特殊结构实现高效双向搜索与对称性破缺。计算结果表明，集合分割模型的线性规划松弛能产生强对偶界，且本文的分支定价方法显著优于现有技术。该方法能够在先前所需时间的极小部分内解决已解实例，并攻克了13个此前未解的实例——其中包含一个76节点的实例，这较之前52节点的求解极限取得了显著突破。