Combinatorial optimization and reasoning with graph neural networks

TL;DR

This survey analyzes how graph neural networks can transform combinatorial optimization by leveraging graph structure to improve primal and dual solving stages, and by enabling algorithmic reasoning that aligns with classical CO methods. It categorizes approaches by learning paradigm (supervised, unsupervised, RL, imitation) and by CO task (primal solution construction, duality/proofs, and algorithmic reasoning). The paper also discusses limitations (expressivity, generalization, inference cost) and proposes directions like speed–accuracy trade-offs, programmatic primitives, perceptive CO, and framework integration. Overall, it highlights a growing toolkit where GNNs augment CO solvers, with potential for end-to-end pipelines that handle raw inputs and real-world data more effectively than traditional abstract formulations.

Abstract

Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks (GNNs), as a key building block for combinatorial tasks, either directly as solvers or by enhancing exact solvers. The inductive bias of GNNs effectively encodes combinatorial and relational input due to their invariance to permutations and awareness of input sparsity. This paper presents a conceptual review of recent key advancements in this emerging field, aiming at optimization and machine learning researchers.

Figures (6)

• Figure 1: A complete graph with edge labels (blue and red) and its optimal solution for the TSP (in green). Blue edges have a cost of $1$ and red edges a cost of $2$.
• Figure 2: Illustration of the neighborhood aggregation step of a GNN around node $v_4$.
• Figure 3: Variable selection in the branch-and-bound integer programming algorithm as a MDP.
• Figure 4: Illustration of algorithmic alignment, in the case of the Bellman-Ford shortest path-finding algorithm Bellman1958. It computes distance estimates for every node, $d_u$, and is shown on the left. Specifically, a GNN aligns well with this dynamic programming update. Node features align with intermediate computed values (red), message functions align with the candidate solutions from each neighbor (blue), and the aggregation function (if, e.g., chosen to be $\max$) aligns with the optimization across neighbors (green).
• Figure 5: The utility of dynamically choosing the graph to reason over for incremental connectivity. It is easy to construct an example path graph (top), wherein deciding whether one vertex is reachable from another requires linearly many GNN iterations. This can be ameliorated by reasoning over different links---namely, ones of the disjoint set union (DSU) data structure Galler1964 that represent each connected component as a rooted tree. At the bottom, from left-to-right, we illustrate the evolution of the DSU for the graph above, once the edge $(h, d)$ is added and query $(b, g)$ executed. Note how the DSU gets compressed after each query Tarjan1975, thus making it far easier for subsequent querying of whether two nodes share the same root.

Theorems & Definitions (8)

• Definition 1: Combinatorial optimization instance
• Example 1: Traveling Salesperson Problem
• Definition 2: Linear programming instance
• Example 2
• Definition 3: SAT
• Example 3
• Definition 4: Constraint satisfaction problem instance
• Example 4