UniHetCO: A Unified Heterogeneous Representation for Multi-Problem Learning in Unsupervised Neural Combinatorial Optimization

A key challenge in developing unified neural solvers for combinatorial optimization (CO) is the efficient generalization of models from a given set of tasks to new tasks unseen during initial training. To address this, we first establish a new GNN encoder, which uses a GCON module as a form of expressive message passing together with energy-based unsupervised loss functions. This model achieves highly competitive performance across multiple CO tasks when trained individually on each task. We then leverage knowledge from the computational reducibility literature to propose pretraining and fine-tuning strategies that transfer effectively (a) between MVC, MIS and MaxClique, and (b) in a multi-task learning setting that additionally incorporates MaxCut, MDS and graph coloring. Additionally, in a leave-one-out, multi-task learning setting, we observe that pretraining on all but one task almost always leads to faster convergence on the remaining task when fine-tuning, while avoiding negative transfer. Our findings indicate that learning common representations across multiple graph CO problems is viable through the use of expressive message passing coupled with pretraining strategies that are informed by the polynomial reducibility literature, thereby taking an important step towards enabling the development of foundational models for neural CO. We provide an open source implementation of our work at https://github.com/semihcanturk/COPT-MT.

Self-supervised learning for combinatorial optimization has emerged as a promising paradigm for solving discrete optimization problems with neural networks, but a central challenge remains: handling hard combinatorial constraints within continuous, gradient-based training. Continuously extending combinatorial objectives to convex domains is a powerful technique, yet existing approaches often require projection steps that constrain neural network outputs to lie inside the feasible polytope and rely on ad-hoc and problem-specific constructions. We propose a general framework in which the neural network is allowed to predict arbitrary continuous vectors that could potentially lie outside of the feasible polytope. These predictions are then approximated by sparse convex combinations of feasible solutions using a geometric decomposition algorithm based on Frank--Wolfe methods and approximate Caratheodory results. This decomposition induces an a.e.-differentiable, self-supervised loss defined as the expected value of the discrete objective. The same procedure provides an automatic rounding guarantee at inference time. We demonstrate strong empirical performance across multiple combinatorial problems, including the Quadratic Assignment Problem, Maximum Coverage, and the Traveling Salesperson Problem.

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.