Self-Improving Neural-Guided Pruning: A Graph Neural Network Framework for Scalable Mixed Bundle Pricing

Jul 16, 2026

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Mixed bundle pricing is a revenue management task focused on selecting product bundles and setting prices to maximize expected profit in domains like e-commerce, tourism, and video games. Exact mixed bundling models are computationally hard because the number of candidate bundles grows exponentially. A graph neural network (GNN)-guided pruning-then-optimization framework models instances as segment–product graphs, predicts inclusion probabilities, prunes the bundle space, and solves the mixed bundling formulation on the retained bundles. Self-improvement uses large-scale generated high-quality solutions as near-optimal labels, enabling scalable learning. Experiments show 13–21% higher profit than bundle-size pricing while using only ~2% of runtime, and theory supports the expressiveness of edge-output learning for optimal assignments.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages52

Reading Time2 h 11 min

Introduction

Background of mixed bundle pricing

Challenges: combinatorial explosion and customer self-selection

Related work and learning to optimize (L2O)

Method Overview

GNN-guided pruning-then-optimization framework

Segment–product graph representation

Self-improvement for scalable supervision

Theory and Expressiveness

Edge-level learning target rationale

Experiments

Profit comparisons and runtime efficiency

arXiv :2509 .22557v 5 [ cs .LG] 9 Jul 2026 Self-Improving Neural-Guided Pruning: A Graph Neural Network Framework for Scalable Mixed Bundle Pricing Liangyu Ding Chenghan Wu Guokai Li Zizhuo Wang School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, China {liangyuding, chenghanwu, [guokaili}@link.cuhk.edu.cn](guokaili}@link.cuhk.edu.cn), [wangzizhuo@cuhk.edu.cn](wangzizhuo@cuhk.edu.cn) Abstract Mixed bundle pricing is a classic revenue management problem arising in industries such as ecommerce, tourism, and video games. It refers to designing product combinations (i.e. , bundles) and determining their prices to maximize expected profit. Exact mixed bundling models capture this structure but become computationally intractable because the number of candidate bundles grows exponentially with the number of products. We develop a graph neural network (GNN)-guided pruning-then-optimization framework for bundle pricing with (non-)additive valuations. The method represents each instance as a compact segment-product graph, predicts segment-product inclusion probabilities, and accordingly prunes the exponential bundle space into a small candidate family; the final prices and bundle offerings are obtained by solving the mixed bundling formulation over the retained bundles, possibly refined by a GNN-guided local search. Because exact labels are available only at small scales, we further propose an iterative self-improvement procedure: the current GNN policies generate high-quality solutions on large-scale instances, which serve as nearoptimal labels for training a stronger model at larger scales. Theoretically, we show that under mild conditions the proposed edge-output GNN class is expressive enough to represent the optimal product-assignment mapping, justifying the edge-level learning target. Numerical experiments show that the fastest proposed policy delivers 13–21% higher profit than bundle-size pricing on instances with up to 100 products at about 2% of its runtime. The framework shows when and why learning creates value in bundle pricing: prediction is used not to replace optimization but to identify where optimization effort should be concentrated, preserving the pricing rigor of exact mixed bundling while breaking its scalability barrier. As product catalogs grow beyond the reach of exact labels, the self-improvement procedure lets the deployed model generate its own retraining supervision, keeping the GNN maintainable at scale. Keywords: bundle pricing; revenue management; graph neural networks; learning to optimize; machine learning. 1 Introduction Bundle pricing is a widely adopted strategy across industries such as e-commerce, digital subscriptions, and retail. It refers to the practice where a firm provides combinations (i.e., “bundles”) of products or services at discounted prices, supplementing the traditional component pricing (CP) strategy where products are only sold separately. For instance, brands like Tula Skincare actively employ a mixed bundling strategy by offering individual items alongside a range of curated sets and starter kit bundles.1 As illustrated in Figure 1, these bundles explicitly highlight both the percentage and absolute dollar savings to clearly communicate value and incentivize larger purchases. In the e-commerce sector, industry reports suggest that KIND Snacks reported a 24% increase in average order value (AOV) after introducing 1 [https://www.ordergroove.com/blog/product-bundling/](https://www.ordergroove.com/blog/product-bundling/) “build-your-own” bundles that empower customers to actively select their own personalized product combinations, while Peet’s Coffee drove a 27% increase in new subscribers within a single year through targeted subscription bundles. To implement this strategy effectively, a firm needs to solve a complex optimization problem: determining which subsets of products to offer and setting their corresponding prices to maximiz

FAQs

What problem does the document address in mixed bundle pricing?

It addresses maximizing expected profit by choosing product bundles and prices under customer self-selection of the option that maximizes each customer’s surplus. The difficulty is that exact mixed bundling models become intractable as the number of products grows.

How does the proposed GNN-guided pruning-then-optimization framework reduce computational cost?

It represents each instance as a compact segment–product graph, predicts inclusion probabilities for bundles, and prunes the exponentially large bundle space into a small candidate family. It then computes optimal prices and bundle offerings by solving the mixed bundling formulation over the retained bundles, optionally refined by local search guided by the GNN.

How does the self-improvement procedure work when exact labels are available only at small scales?

The current GNN produces high-quality solutions on large-scale instances, which serve as near-optimal labels for training a stronger model at larger scales. This allows the deployed model to generate its own retraining supervision while staying maintainable at scale.

Jul 16, 2026

Upload by Himbo