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

Abstract

Unsupervised neural combinatorial optimization (NCO) offers an appealing alternative to supervised approaches by training learning-based solvers without ground-truth solutions, directly minimizing instance objectives and constraint violations. Yet for graph node subset-selection problems (e.g., Maximum Clique and Maximum Independent Set), existing unsupervised methods are typically specialized to a single problem class and rely on problem-specific surrogate losses, which hinders learning across classes within a unified framework. In this work, we propose UniHetCO, a unified heterogeneous graph representation for constrained quadratic programming-based combinatorial optimization that encodes problem structure, objective terms, and linear constraints in a single input. This formulation enables training a single model across multiple problem classes with a unified label-free objective. To improve stability under multi-problem learning, we employ a gradient-norm-based dynamic weighting scheme that alleviates gradient imbalance among classes. Experiments on multiple datasets and four constrained problem classes demonstrate competitive performance with state-of-the-art unsupervised NCO baselines, strong cross-problem adaptation potential, and effective warm starts for a commercial classical solver under tight time limits.

Figures (4)

• Figure 1: High-level comparison between existing single-problem (top) and our multi-problem (bottom) neural combinatorial optimization framework. By encoding objectives and constraints into the input graph, our approach enables joint training across multiple problem classes.
• Figure 2: Examples of MIS, MC, MVC and MDS instances encoded in our unified heterogeneous graph. The objectives and constraints are listed for each problem class, along with a simple example for $\mathbf{Q}, \mathbf{c}, \mathbf{A},$ and $\mathbf{b}$ and the corresponding heterogeneous graph defined in Sec. \ref{['subsec:unified-repr']}. Red nodes are decision variables; black edges are original-graph relations; green edges are objective-graph relations. Blue squares are constraint nodes, connected to variables by blue dashed edges.
• Figure 3: Results on cross-problem generalization and adaptation on the Twitter dataset across all four problem classes. In each subfigure, the target class is held out during training (e.g., MIS is evaluated after training on MC, MVC, and MDS), and performance is reported under zero-shot transfer and fine-tuning. The dashed horizontal line denotes the UniHetCO-Single baseline approximation ratio. Note that it is higher better for MIS and MC, and lower better for MVC and MDS.
• Figure 4: Results on warm-start for Gurobi ($\leq$ 0.2s) for the MC problem. We use three datasets Twitter, RB200 and SparseSuit. UniHetCO-Single yields the strongest warm-start, with UniHetCO-DW offering smaller but still positive gains.