Reinforcement Learning-Based Formulations With Hamiltonian-Inspired Loss Functions for Combinatorial Optimization Over Graphs

Abstract

Quadratic Unconstrained Binary Optimization (QUBO) is a versatile approach used to represent a wide range of NP-hard Combinatorial Optimization (CO) problems through binary variables. The transformation of QUBO to an Ising Hamiltonian is recognized as an effective method for solving key optimization problems using quantum algorithms. Recently, PI-GNN, a generic framework, has been proposed to address CO problems over graphs based on QUBO with Hamiltonian loss function to train the underlying GNN architecture. Though PI-GNN is highly scalable, it exhibits a noticeable decrease in terms of the number of satisfied constraints with higher graph densities. In this paper, firstly, we identify the limitations and empirically present our strategy to improve PI-GNN’s performance. Secondly, we formulate and evaluate two strategies to integrate QUBO-Hamiltonian as the generic loss function in Reinforcement learning-based (RL) frameworks. The major contribution of our work lies in understanding the feasibility and quality of the QUBO-based generic reward function in an unsupervised RL setup in addressing graph-based CO problems. Empirically, through our empirical evaluation (Our implementation can be found in https://tinyurl.com/5apnymz7), we have observed up to 44% improvement in terms of the number of satisfied constraints over PI-GNN in a representative Max-Cut problem.

Keywords

• Deep reinforcement learning
• graph neural network
• Hamiltonian function
• Monte Carlo tree search