BQ-NCO: Bisimulation Quotienting for Generalizable Neural Combinatorial Optimization

Despite the success of Neural Combinatorial Optimization methods for end-to-end heuristic learning, out-of-distribution generalization remains a challenge. In this paper, we present a novel formulation of combinatorial optimization (CO) problems as Markov Decision Processes (MDPs) that effectively leverages symmetries of the CO problems to improve out-of-distribution robustness. Starting from the standard MDP formulation of constructive heuristics, we introduce a generic transformation based on bisimulation quotienting (BQ) in MDPs. This transformation allows to reduce the state space by accounting for the intrinsic symmetries of the CO problem and facilitates the MDP solving. We illustrate our approach on the Traveling Salesman, Capacitated Vehicle Routing and Knapsack Problems. We present a BQ reformulation of these problems and introduce a simple attention-based policy network that we train by imitation of (near) optimal solutions for small instances from a single distribution. We obtain new state-of-the-art generalization results for instances with up to 1000 nodes from synthetic and realistic benchmarks that vary both in size and node distributions.

On the Generalization of Neural Combinatorial Optimization Heuristics

Neural Combinatorial Optimization approaches have recently leveraged the expressiveness and flexibility of deep neural networks to learn efficient heuristics for hard Combinatorial Optimization (CO) problems. However, most of the current methods lack generalization: for a given CO problem, heuristics which are trained on instances with certain characteristics underperform when tested on instances with different characteristics. While some previous works have focused on varying the training instances properties, we postulate that a one-size-fit-all model is out of reach. Instead, we formalize solving a CO problem over a given instance distribution as a separate learning task and investigate meta-learning techniques to learn a model on a variety of tasks, in order to optimize its capacity to adapt to new tasks. Through extensive experiments, on two CO problems, using both synthetic and realistic instances, we show that our proposed meta-learning approach significantly improves the generalization of two state-of-the-art models.