Constant-factor approximation for general choice models

Determine whether a constant-factor approximation exists for the joint inventory placement, assortment personalization, and order fulfillment problem under a general customer choice model when a polynomial-time assortment optimization oracle is available, thereby bridging the gap between the paper’s asymptotic approximation results and its constant-factor results for the multinomial logit choice model.

Background

The paper develops asymptotically optimal approximation algorithms for the joint inventory placement, assortment personalization, and order fulfillment problem under general choice models, with guarantees improving as warehouse capacities and product supplies increase. For the multinomial logit choice model, it establishes constant-factor approximation guarantees using a DR-submodular surrogate and submodular maximization.

For general choice models, however, the paper does not obtain a constant approximation factor that is independent of inventory capacities and product supplies. The authors explicitly identify the existence of such a guarantee, under the assumption that single-period assortment optimization can be solved by a polynomial-time oracle, as an open question that would connect the general-choice-model and multinomial-logit results.

References

To apply the same architecture to another leaf model, intermediate solver iterates must provide valid lower bounds, and those bounds must remain transferable across nested and disjoint samples. We establish these properties for constrained MNL leaves. Whether richer choice models share them remains to be studied.

— Learning Choice Model Trees for Feature-Based Multi-Product Pricing: Exact Optimization and Field Evidence  (2609.16952 - Zhang et al., 15 Sep 2026) in Section Conclusions

Second, a significant open question is whether a constant-factor approximation exists for a general choice model under assortment oracle assumption, bridging the gap between our asymptotic results and the MNL case.

— Joint Inventory Placement, Assortment Personalization, and Order Fulfillment for Substitutable Products  (2608.14275 - Fadin et al., 14 Aug 2026) in Section 8, Conclusion