DistrictNet: Decision-aware learning for geographical districting

TL;DR

DistrictNet addresses large-scale geographical districting with uncertain demand by learning a structured surrogate: predicting edge costs for a capacitated minimum spanning tree (CMST) via a graph neural network and solving the CMST as a surrogate to the original districting problem. The model is trained in a decision-aware fashion using a Fenchel-Young loss and a randomized target-construction to propagate gradients through the CMST layer, enabling efficient inference on large instances. Empirical results on real-world UK and French cities show DistrictNet achieves substantial cost reductions (around 10% on average) and better district compactness, with strong generalization to out-of-distribution sizes and city structures. The approach integrates an optimization layer with deep learning to capture the combinatorial structure of graph partitioning, offering a generic methodology for related geographical partitioning tasks beyond districting and routing.

Abstract

Districting is a complex combinatorial problem that consists in partitioning a geographical area into small districts. In logistics, it is a major strategic decision determining operating costs for several years. Solving districting problems using traditional methods is intractable even for small geographical areas and existing heuristics often provide sub-optimal results. We present a structured learning approach to find high-quality solutions to real-world districting problems in a few minutes. It is based on integrating a combinatorial optimization layer, the capacitated minimum spanning tree problem, into a graph neural network architecture. To train this pipeline in a decision-aware fashion, we show how to construct target solutions embedded in a suitable space and learn from target solutions. Experiments show that our approach outperforms existing methods as it can significantly reduce costs on real-world cities.

Figures (9)

• Figure 1: DistrictNet solves a complex districting problem by parameterizing and solving a CMST. The GNN $\phi$ predicts a vector of edge weights $\theta$ based on the covariates of the instance $x$. These edge weights parameterize a CMST, which is solved using a black-box combinatorial solver. The CMST solution $\hat{y}$ is finally converted into a districting solution $\hat{\lambda}$. Training this pipeline in a decision-aware manner requires propagating a loss gradient back to the GNN.
• Figure 2: Districting solutions given by BD, FIG, PredictGNN, and DistrictNet for the city of Manchester with district target sizes of $20$ BUs. The depot is shown as a white star.
• Figure 3: Cost relative to DistrictNet for target district size $t=20$ and varying city size.
• Figure 4: Distribution of district cost for varying target size.
• Figure 5: Relative cost of DistrictNet with increasing data.