AI 中文总结
研究长度受限循环划分问题，将其建模为集合划分模型，采用基于动态规划的分支定价切割算法求解。该算法利用问题结构高效搜索并打破对称，改进显著，能快速解决已解实例，还解决多个未解实例，突破节点数量限制。
AI 中文摘要
长度受限循环划分问题（LCCP）是一个图优化问题，要将一组节点划分为最少数量的循环。每个节点都关联一个关键时间，每个循环的长度不得超过循环中任何节点的关键时间。我们将LCCP 表述为集合划分模型，并使用精确的分支定价切割方法求解。基于动态规划的定价算法利用定价问题的特殊结构进行高效双向搜索和打破对称性来生成改进循环。计算结果表明，集合划分模型的线性规划松弛产生了非常强的对偶界，我们的分支定价切割方法比现有技术有显著改进。它能在更短时间内解决之前已解实例，并以数值安全界解决 14 个之前未解实例，其中一个有 76 个节点，远超之前 52 个节点的限制。
英文摘要
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-price-and-cut approach. Our dynamic programming-based pricing algorithm to generate improving cycles exploits 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 very strong dual bounds and our branch-price-and-cut method improves significantly over the state of the art. It is able to solve previously solved instances in a fraction of the time and closes 14 previously unsolved instances with numerically safe bounds, one of which has 76 nodes, a notable improvement over the previous limit of 52 nodes.
CommentsarXiv admin note: text overlap with arXiv:2401.17937
