A Numerically-safe Branch-Price-and-Cut Algorithm for the Length-Constrained Cycle Partition Problem

2026-07-17

AI Summary

This study addresses the Length-Constrained Cycle Partition problem (LCCP), which seeks to partition the nodes of a graph into the minimum number of cycles such that the length of each cycle does not exceed the critical time associated with its nodes. Building upon a set-partitioning formulation, the authors propose a numerically stable branch-price-and-cut framework that integrates a dynamic programming-based pricing algorithm, bidirectional search, and symmetry-breaking strategies to efficiently generate improving cycles and accelerate convergence. The approach yields exceptionally strong linear programming dual bounds, substantially outperforming existing methods. It successfully closes 14 large-scale benchmark instances that had remained open for years and extends the solvable problem size from 52 to 76 nodes.

Abstract

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.

Problem

• length-constrained cycle partition
• graph optimization
• set partitioning
• critical time
• cycle length

Innovation

• branch-price-and-cut
• length-constrained cycle partition
• dynamic programming pricing
• numerically safe bounds
• symmetry breaking