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.