Assortment Optimization under the Mallows Model: Independent-Utility Choice Equivalence and Its Algorithmic Implications
Abstract: The Mallows model is a well-established construct in preference modeling, describing random rankings through their Kendall-Tau distance from a reference permutation. Its compact parameterization and interpretable treatment of preference uncertainty have made it influential across statistics, machine learning, probability, and revenue management. In the latter domain, recent literature has studied the embedding of Mallows-driven choice into assortment optimization. Early approximation schemes in this context require stylized structural simplifications or restrictive parametric assumptions. In the absence of these levers, a highly involved quasi-polynomial-time approximation scheme has subsequently been proposed, leaving wide open the question of whether near-optimal assortments can be computed in true polynomial time. The main contribution of this focused technical paper resides in closing these gaps by uncovering a striking independent-utility structure within Mallows choices, and turning this representation into a polynomial-time approximation scheme. Our central structural characterization explicitly constructs a single collection of mutually independent utilities, reproducing Mallows choice probabilities simultaneously for all offered sets. This exact choice equivalence connects pending Mallows-related challenges to the recent approximation framework of Bowers et al. (FOCS 2025) for assortment optimization under independent random utilities. We make this connection algorithmically feasible through a finite-support reduction that produces distributions of polynomial size while controlling revenue loss uniformly over all assortments. Fused together, these ingredients lead to an $O(n^{\tilde{O}(1/ε^2)})$-time approximation scheme for assortment optimization under the Mallows model, while allowing arbitrary product prices, concentration parameters, and modal rankings.