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：长度约束圈划分问题（LCCP）是一类图优化问题，要求将节点集合划分为数量最少的圈。每个节点关联一个临界时间，且每个圈的长度不得超过该圈中任意节点的临界时间。我们将LCCP建模为集合划分模型，并采用精确的分支定价方法求解。我们使用基于动态规划的定价算法生成改进圈，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，集合划分模型的线性规划松弛能产生强对偶界，且我们的分支定价方法显著优于现有技术。该方法能在以往所需时间的一小部分内求解已闭实例，并解决了13个此前未解的实例——其中包含76个节点的实例，相较于此前52个节点的求解极限实现了显著提升。