Quasi-polynomial time approximation schemes for assortment optimization under Mallows-based rankings
In spite of its widespread applicability in learning theory, probability, combinatorics, and in various other fields, the Mallows model has only recently been examined from revenue management perspectives. To our knowledge, the only provably-good approaches for assortment optimization under the Mall...
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| Published in: | Mathematical programming Vol. 208; no. 1-2; pp. 111 - 171 |
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| Abstract | In spite of its widespread applicability in learning theory, probability, combinatorics, and in various other fields, the Mallows model has only recently been examined from revenue management perspectives. To our knowledge, the only provably-good approaches for assortment optimization under the Mallows model have recently been proposed by Désir et al. (Oper Res 69(4):1206–1227, 2021), who devised three approximation schemes that operate in very specific circumstances. Unfortunately, these algorithmic results suffer from two major limitations, either crucially relying on strong structural assumptions, or incurring running times that exponentially scale either with the ratio between the extremal prices or with the Mallows concentration parameter. The main contribution of this paper consists in devising a quasi-polynomial-time approximation scheme for the assortment optimization problem under the Mallows model in its utmost generality. In other words, for any accuracy level
ϵ
>
0
, our algorithm identifies an assortment whose expected revenue is within factor
1
-
ϵ
of optimal, without resorting to any structural or parametric assumption whatsoever. Our work sheds light on newly-gained structural insights surrounding near-optimal Mallows-based assortments and fleshes out some of their unexpected algorithmic consequences. |
|---|---|
| AbstractList | In spite of its widespread applicability in learning theory, probability, combinatorics, and in various other fields, the Mallows model has only recently been examined from revenue management perspectives. To our knowledge, the only provably-good approaches for assortment optimization under the Mallows model have recently been proposed by Désir et al. (Oper Res 69(4):1206-1227, 2021), who devised three approximation schemes that operate in very specific circumstances. Unfortunately, these algorithmic results suffer from two major limitations, either crucially relying on strong structural assumptions, or incurring running times that exponentially scale either with the ratio between the extremal prices or with the Mallows concentration parameter. The main contribution of this paper consists in devising a quasi-polynomial-time approximation scheme for the assortment optimization problem under the Mallows model in its utmost generality. In other words, for any accuracy level [Formula omitted], our algorithm identifies an assortment whose expected revenue is within factor [Formula omitted] of optimal, without resorting to any structural or parametric assumption whatsoever. Our work sheds light on newly-gained structural insights surrounding near-optimal Mallows-based assortments and fleshes out some of their unexpected algorithmic consequences. In spite of its widespread applicability in learning theory, probability, combinatorics, and in various other fields, the Mallows model has only recently been examined from revenue management perspectives. To our knowledge, the only provably-good approaches for assortment optimization under the Mallows model have recently been proposed by Désir et al. (Oper Res 69(4):1206–1227, 2021), who devised three approximation schemes that operate in very specific circumstances. Unfortunately, these algorithmic results suffer from two major limitations, either crucially relying on strong structural assumptions, or incurring running times that exponentially scale either with the ratio between the extremal prices or with the Mallows concentration parameter. The main contribution of this paper consists in devising a quasi-polynomial-time approximation scheme for the assortment optimization problem under the Mallows model in its utmost generality. In other words, for any accuracy level ϵ > 0 , our algorithm identifies an assortment whose expected revenue is within factor 1 - ϵ of optimal, without resorting to any structural or parametric assumption whatsoever. Our work sheds light on newly-gained structural insights surrounding near-optimal Mallows-based assortments and fleshes out some of their unexpected algorithmic consequences. |
| Audience | Academic |
| Author | Segev, Danny Rieger, Alon |
| Author_xml | – sequence: 1 givenname: Alon surname: Rieger fullname: Rieger, Alon organization: Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University – sequence: 2 givenname: Danny orcidid: 0000-0003-4684-2185 surname: Segev fullname: Segev, Danny email: segevdanny@tauex.tau.ac.il organization: Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University |
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| Keywords | Mallows model Assortment optimization 90C39 Dynamic programming 68W25 90C27 Quasi-PTAS |
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| References_xml | – reference: ŞimşekASTopalogluHTechnical note: an expectation-maximization algorithm to estimate the parameters of the Markov chain choice modelOper. Res.20186637487603826853 – reference: JonesBWeighted games of best choiceSIAM J. Discret. Math.20203413994144062797 – reference: BlanchetJHGallegoGGoyalVA Markov chain approximation to choice modelingOper. Res.20166448869053532860 – reference: AouadAFariasVLeviRSegevDThe approximability of assortment optimization under ranking preferencesOper. Res.2018666166116693898852 – reference: Liu, A., Moitra, A.: Efficiently learning mixtures of Mallows models. In: Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, pp. 627–638 (2018) – reference: DésirAGoyalVSegevDYeCCapacity constrained assortment optimization under the Markov chain based choice modelManag. Sci.2020662698721 – reference: BrontJJMMéndez-DiazIVulcanoGA column generation algorithm for choice-based network revenue managementOper. 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| Title | Quasi-polynomial time approximation schemes for assortment optimization under Mallows-based rankings |
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