Reward-Weighted Preference Optimization for Selective Combinatorial Optimization Problems

Overview & Synopsis

Reinforcement learning has emerged as an effective approach for combinatorial optimization, but most existing methods focus on problems such as the Travelling Salesperson Problem (TSP) in which all elements must be served. Selective combinatorial optimization problems (SCOPs), including the Orienteering Problem (OP), Prize-Collecting TSP (PCTSP), Stochastic PCTSP and Knapsack Problem (KP), additionally require selecting which elements to include. This selection creates sampled solutions with substantially different objective values, yet standard preference optimization reduces such comparisons to binary rankings and discards the magnitude of these differences. We introduce Reward-Weighted Preference Optimization (RWPO), a pairwise training objective that incorporates reward-gap information into preference learning. For candidate trajectories sampled from the same instance, RWPO weights each preference loss according to the normalized difference in their rewards, assigning greater training emphasis to comparisons that represent larger improvements in solution quality. We further combine RWPO with scale-specific fine-tuning and reward- and likelihood-weighted tilted sampling to adapt trained policies to larger and shifted instance distributions without relying on local-search post-processing. Experiments on the OP, PCTSP, Stochastic PCTSP, and KP show that RWPO consistently improves over unweighted preference optimization and policy-gradient baselines across a wide range of instance sizes. On the OP, the resulting solver exceeds the mean reward of the Compass metaheuristic at scales from 100 to 2,000 nodes. The results demonstrate that preserving reward-gap information can substantially improve preference-based learning for selective combinatorial optimization and support generalization to large-scale instances.