A Graph Neural Network-Based QUBO-Formulated Hamiltonian-Inspired Loss Function for Combinatorial Optimization using Reinforcement Learning

Quadratic Unconstrained Binary Optimization (QUBO) is a generic technique to model various NP-hard Combinatorial Optimization problems (CO) in the form of binary variables. Ising Hamiltonian is used to model the energy function of a system. QUBO to Ising Hamiltonian is regarded as a technique to solve various canonical optimization problems through quantum optimization algorithms. Recently, PI-GNN, a generic framework, has been proposed to address CO problems over graphs based on Graph Neural Network (GNN) architecture. They introduced a generic QUBO-formulated Hamiltonian-inspired loss function that was directly optimized using GNN. PI-GNN is highly scalable but there lies a noticeable decrease in the number of satisfied constraints when compared to problem-specific algorithms and becomes more pronounced with increased graph densities. Here, We identify a behavioral pattern related to it and devise strategies to improve its performance. Another group of literature uses Reinforcement learning (RL) to solve the aforementioned NP-hard problems using problem-specific reward functions. In this work, we also focus on creating a bridge between the RL-based solutions and the QUBO-formulated Hamiltonian. We formulate and empirically evaluate the compatibility of the QUBO-formulated Hamiltonian as the generic reward function in the RL-based paradigm in the form of rewards. Furthermore, we also introduce a novel Monty Carlo Tree Search-based strategy with GNN where we apply a guided search through manual perturbation of node labels during training. We empirically evaluated our methods and observed up to 44% improvement in the number of constraint violations compared to the PI-GNN.

Understanding the Usage of QUBO-based Hamiltonian Function in Combinatorial Optimization over Graphs: A Discussion Using Max Cut (MC) Problem

Quadratic Unconstrained Binary Optimization (QUBO) is a generic technique to model various NP-hard combinatorial optimization problems in the form of binary variables. The Hamiltonian function is often used to formulate QUBO problems where it is used as the objective function in the context of optimization. In this study, we investigate how reinforcement learning-based (RL) paradigms with the presence of the Hamiltonian function can address combinatorial optimization problems over graphs in QUBO formulations. We use Graph Neural Network (GNN) as the message-passing architecture to convey the information among the nodes. We have centered our discussion on QUBO formulated Max-Cut problem but the intuitions can be extended to any QUBO supported canonical NP-Hard combinatorial optimization problems. We mainly investigate three formulations, Monty-Carlo Tree Search with GNN-based RL (MCTS-GNN), DQN with GNN-based RL, and a generic GNN with attention-based RL (GRL). Our findings state that in the RL-based paradigm, the Hamiltonian function-based optimization in QUBO formulation brings model convergence and can be used as a generic reward function. We also analyze and present the performance of our RL-based setups through experimenting over graphs of different densities and compare them with a simple GNN-based setup in the light of constraint violation, learning stability and computation cost. As per one of our findings, all the architectures provide a very comparable performance in sparse graphs as per the number of constraint violation whreas MCTS-GNN gives the best performance. In the similar criteria, the performance significantly starts to drop both for GRL and simple GNN-based setups whereas MCTS-GNN and DQN shines. We also present the corresponding mathematical formulations and in-depth discussion of the observed characteristics during experimentations.