Branch and Price for the Length-Constrained Cycle Partition Problem

TL;DR

This work tackles the length-constrained cycle partition problem (LCCP), where the node set $V$ must be partitioned into disjoint cycles such that $t(C) \le q(C)$ for every cycle, with $t(C)=\sum_{k=1}^K t_{i_{k-1},i_k}$ and $q(C)=\min_{i\in C} q_i$. It introduces a set-partitioning formulation over all length-feasible cycles and solves the LP via column generation within an exact branch-and-price framework, employing a dynamic programming pricing subroutine for the length-constrained prize-collecting cycle problem (LCPCCP). Key contributions include a DP-based pricing algorithm with dominance pruning and bidirectional search, symmetry breaking, parallel and heuristic pricing, early branching, Farkas pricing, and triangle-inequality exploitation to improve convergence. Computational results on standard benchmarks demonstrate that the approach significantly outperforms the previous state of the art, solving 13 open instances and scaling to instances with up to 76 nodes while yielding tight LP bounds. The findings highlight the effectiveness of the improvement techniques and suggest potential applicability to related problems such as kidney exchange and periodic surveillance planning.

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-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.

Figures (3)

• Figure 1: Example of merging two partial cycles.
• Figure 1: Aggregated results.
• Figure 2: Instances solved over time.

Theorems & Definitions (5)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma