arXiv subjects

Boris Epstein

Publications and source records attributed to Boris Epstein.

4 recordsLinked to original sources

arXiv2024

Optimizing Inventory Placement for a Downstream Online Matching Problem

We study the inventory placement problem of splitting $Q$ units of a single item across warehouses in advance of a downstream online matching problem that represents the dynamic fulfillment decisions of an e-commerce retailer. This is a challenging problem both theoretically, due to the computational complexity of the downstream matching problem, and practically, as the fulfillment team continuously updates its algorithm while the placement team lacks direct evaluation of placement decisions. We compare the performance of three placement procedures based on optimizing surrogate functions that have been studied and applied: Offline, Myopic, and Fluid placement. On the theory side, we show that optimizing inventory placement for the Offline surrogate leads to an $\alpha (1-(1-1/d)^d)$-approximation for the joint placement and fulfillment problem under any demand model that admits an $\alpha$-competitive fulfillment policy. We assume $d$ is an upper bound on how many warehouses can serve any demand location. The crux of our theoretical contribution is to use randomized rounding to derive a tight $(1-(1-1/d)^d)$-approximation for the integer programming problem of optimizing the Offline surrogate. We further show how to extend this result to a multi-SKU setting, improving upon the best known approximation of $1/2$. We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, indeed has vanishing loss. On the experimental side, we evaluate how different combinations of placement and fulfillment procedures perform on a wide array of synthetic instances. When coupled with a good fulfillment procedure, optimizing the Offline surrogate performs best even compared to computationally-intensive simulation procedures, corroborating our theory.

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arXiv2022

Selection and Ordering Policies for Hiring Pipelines via Linear Programming

Motivated by hiring pipelines, we study three selection and ordering problems in which applicants for a finite set of positions must be interviewed or sent offers. There is a finite time budget for interviewing/sending offers, and every interview/offer is followed by a stochastic realization of discovering the applicant's quality or acceptance decision, leading to computationally challenging problems. In the first problem, we study sequential interviewing and show that a computationally tractable, non-adaptive policy that must make offers immediately after interviewing is near-optimal, assuming offers are always accepted. We further show how to use this policy as a subroutine for obtaining a PTAS. In the second problem, we assume that applicants have already been interviewed but only accept offers with some probability; we develop a computationally tractable policy that makes offers for the different positions in parallel, which can be used even if positions are heterogeneous, and is near-optimal relative to a policy that can make the same total number of offers one by one. In the third problem, we introduce a parsimonious model of overbooking where all offers must be sent simultaneously and a linear penalty is incurred for each acceptance beyond the number of positions; we provide nearly tight bounds on the performance of practically motivated value-ordered policies. All in all, our paper takes a unified approach to three different hiring problems, based on linear programming. Our results in the first two problems generalize and improve the existing guarantees due to Purohit et al. (2019) that were between 1/8 and 1/2 to new guarantees that are at least 1-1/e. We also numerically compare three different settings of making offers to candidates (sequentially, in parallel, or simultaneously), providing insight into when a firm should favor each one.

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arXiv2020

Sample-driven optimal stopping: From the secretary problem to the i.i.d. prophet inequality

We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of $N$ items independently with probability $p$, and can observe the relative rankings of these sampled items. Then, the DM faces the remaining items in an online fashion, observing the relative rankings of all revealed items. While scanning the sequence the DM makes irrevocable stop/continue decisions and her reward for stopping the sequence facing the item with rank $i$ is $Y_i$. The goal of the DM is to maximize her reward. We start by studying the case in which the values $Y_i$ are known to the DM, and then move to the case in which these values are adversarial. For the former case, we write the natural linear program that captures the performance of an algorithm, and take its continuous limit. We prove a structural result about this continuous limit, which allows us to reduce the problem to a relatively simple real optimization problem. We establish that the optimal algorithm is given by a sequence of thresholds $t_1\le t_2\le\cdots$ such that the DM should stop if seeing an item with current ranking $i$ after time $t_i$. Additionally we are able to recover several classic results in the area such as those for secretary problem and the minimum ranking problem. For the adversarial case, we obtain a similar linear program with an additional stochastic dominance constraint. Using the same machinery we are able to pin down the optimal competitive ratios for all values of $p$. Notably, we prove that as $p$ approaches 1, our guarantee converges linearly to 0.745, matching that of the i.i.d.~prophet inequality. Also interesting is the case $p=1/2$, where our bound evaluates to $0.671$, which improves upon the state of the art.

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arXiv2019

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given $n$ cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on $n$ consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, as in the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least $0.45292$. In the second one, similar to the prophet inequality, the player maximizes the expectation of the selected hidden value. We show a guarantee of at least $0.63518$ with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial $n$ visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently $n$ visible numbers in the table. And the third one is similar to the latter but it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. Our guarantee improves upon $1-1/e$ for this problem, which is the currently best known guarantee and only works for the i.i.d. case.

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