arXiv subjects

Danny Segev

Publications and source records attributed to Danny Segev.

At least 19 recordsLinked to original sources

arXiv2026

Resource-Constrained Joint Replenishment via Power-of-$m^{1/k}$ Policies

The continuous-time joint replenishment problem has long served as a foundational inventory management model. Even though its unconstrained setting has seen recent algorithmic advances, the incorporation of resource constraints into this domain precludes the application of newly discovered synchronization techniques. Such constraints arise in a broad spectrum of practical environments where resource consumption is bounded as an aggregate rate over time. However, for nearly four decades, the prevailing approximation guarantee for resource-constrained joint replenishment has remained $\frac{ 1 }{ \ln 2 } \approx 1.4427$, achieved via classical power-of-$2$ policies. In this paper, we circumvent these structural policy restrictions by devising generalized rounding frameworks, demonstrating that a well-known convex relaxation is much tighter than previously established. In particular, we expand our analytical scope to encompass fractional base expansion factors, randomized shifting, and staggered interleaved grids. Through this multifaceted methodology, we present a sequence of gradually improving performance guarantees. First, by proposing a best-of-two framework that exploits structural asymmetries between deterministic power-of-$m^{1/k}$ policies, we surpass the classical barrier to obtain a $1.3776$-approximation. Second, by injecting a random shift into the logarithmic grid domain and formulating a factor-revealing linear program to optimize a dual-policy approach, we attain a $1.2512$-approximation. Finally, by superimposing a secondary offset grid to subdivide rounding intervals and suppress holding cost inflation, we utilize interleaved policies to arrive at our ultimate approximation ratio of $\frac{5}{6\ln 2} \approx 1.2023$, which is proven to be best-possible for the class of interleaved power-of-$m^{1/k}$ policies.

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arXiv2026

Economic Warehouse Lot Scheduling: Approximation Schemes via Efficiently-Representable DP-Encoded Policies

In this focused technical paper, we present long-awaited algorithmic advances toward the efficient construction of near-optimal replenishment policies for a true inventory management classic, the economic warehouse lot scheduling problem. While this paradigm has accumulated a massive body of surrounding literature since its inception in the late '50s, we are still very much in the dark as far as basic computational questions are concerned, perhaps due to the intrinsic complexity of dynamic policies in this context. The latter feature forced earlier attempts to either study highly-structured classes of policies or to forgo provably-good performance guarantees altogether; to this day, rigorously analyzable results have been few and far between. The current paper develops novel analytical foundations for directly competing against dynamic policies. Combined with further algorithmic progress and newly-gained insights, these ideas culminate in a polynomial-time approximation scheme for constantly-many commodities. In this regard, the efficient design of $\epsilon$-optimal dynamic policies appeared to have been out of reach, since beyond their inherent algorithmic challenges, even the polynomial-space representation of such policies has been a fundamental open question.

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arXiv2026

Economic Warehouse Lot Scheduling: Breaking the 2-Approximation Barrier

The economic warehouse lot scheduling problem is a foundational inventory-theory model, capturing computational challenges in dynamically coordinating replenishment decisions for multiple commodities subject to a shared capacity constraint. Even though this model has generated a vast body of literature over the last six decades, our algorithmic understanding has remained surprisingly limited. Indeed, for general problem instances, the best-known approximation guarantees have remained at a factor of $2$ since the mid-1990s. These guarantees were attained by the now-classic work of Anily [Operations Research, 1991] and Gallego, Queyranne, and Simchi-Levi [Operations Research, 1996] via the highly-structured class of "stationary order sizes and stationary intervals" (SOSI) policies, thereby avoiding direct competition against fully dynamic policies. The main contribution of this paper resides in developing new analytical foundations and algorithmic techniques that enable such direct comparisons, leading to the first provable improvement over the $2$-approximation barrier. Leveraging these ideas, we design a constructive approach that allows us to balance cost and capacity at a finer granularity than previously possible via SOSI-based methods. Consequently, given any economic warehouse lot scheduling instance, we present a polynomial-time construction of a random capacity-feasible dynamic policy whose expected long-run average cost is within factor $2-\frac{17}{5000} + \epsilon$ of optimal.

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arXiv2026

Approximation Schemes for Sequential Hiring Problems

The main contribution of this paper resides in providing novel algorithmic advances and analytical insights for the sequential hiring problem, a recently introduced dynamic optimization model where a firm adaptively fills a limited number of positions from a pool of applicants with known values and acceptance probabilities. While earlier research established a strong foundation -- notably an LP-based $(1 - \frac{e^{-k}k^k}{k!})$-approximation by Epstein and Ma (Operations Research, 2024) -- the attainability of superior approximation guarantees has remained a central open question. Our work addresses this challenge by establishing the first polynomial-time approximation scheme for sequential hiring, proposing an $O(n^{O(1)} \cdot T^{2^{\tilde{O}(1/\epsilon^{2})}})$-time construction of semi-adaptive policies whose expected reward is within factor $1 - \epsilon$ of optimal. To overcome the constant-factor optimality loss inherent to earlier literature, and to circumvent intrinsic representational barriers of adaptive policies, our approach is driven by the following innovations: -- The block-responsive paradigm: We introduce block-responsive policies, a new class of decision-making strategies, selecting ordered sets (blocks) of applicants rather than single individuals, while still allowing for internal reactivity. -- Adaptivity and efficiency: We prove that these policies can nearly match the performance of general adaptive policies while utilizing polynomially-sized decision trees. -- Efficient construction: By developing a recursive enumeration-based framework, we resolve the problematic ``few-positions'' regime, bypassing a fundamental hurdle that hindered previous approaches.

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arXiv2025

Approximation Algorithms for Fair Repetitive Scheduling

We consider a recently introduced fair repetitive scheduling problem involving a set of clients, each asking for their associated job to be daily scheduled on a single machine across a finite planning horizon. The goal is to determine a job processing permutation for each day, aiming to minimize the maximum total completion time experienced by any client. This problem is known to be NP-hard for quite restrictive settings, with previous work offering exact solution methods for highly-structured special cases. In this paper, we focus on the design of approximation algorithms with provable performance guarantees. Our main contributions can be briefly summarized as follows: (i) When job processing times are day-dependent, we devise a polynomial-time LP-based $2$-approximation, as well as a polynomial-time approximation scheme for a constant number of days. (ii) With day-invariant processing times, we obtain a surprisingly simple $(\frac{1+\sqrt{2}}{2}+\epsilon)$-approximation in polynomial time. This setting is also shown to admit a quasi-polynomial-time approximation scheme for an arbitrary number of days. The key technical component driving our approximation schemes is a novel batching technique, where jobs are conceptually grouped into batches, subsequently leading either to a low-dimensional dynamic program or to a compact configuration LP. Concurrently, while developing our constant-factor approximations, we propose a host of lower-bounding mechanisms that may be of broader interest.

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arXiv2025

Improved Approximation Guarantees and Hardness Results for MNL-Driven Product Ranking

In this paper, we address open computational questions regarding the market share ranking problem, recently introduced by Derakhshan et al. (2022). Their modelling framework incorporates the extremely popular Multinomial Logit (MNL) choice model, along with a novel search-based consider-then-choose paradigm. In a nutshell, the authors devised a Pandora's-Box-type search model, where different customer segments sequentially screen through a ranked list of products, one position after the other, forming their consideration set by including all products viewed up until terminating their inspection procedure. Subsequently, a purchasing decision out of this set is made based on a joint MNL choice model. Our main contribution consists in devising a polynomial-time approximation scheme for the market share ranking problem, utilizing fresh technical developments and analytical ideas, in conjunction with revising the original insights of Derakhshan et al. (2022). Along the way, we introduce a black-box reduction, mapping general instances of the market share ranking problem into ``bounded ratio'' instances, showing that this result directly leads to an elegant and easily-implementable quasi-PTAS. Finally, to provide a complete computational characterization, we prove that the market share ranking problem is strongly $\mathrm{NP}$-hard.

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arXiv2025

Near-Optimal Dynamic Policies for Joint Replenishment in Continuous/Discrete Time

While dynamic policies have historically formed the foundation of most influential papers dedicated to the joint replenishment problem, we are still facing profound gaps in our structural understanding of optimal such policies as well as in their surrounding computational questions. To date, the seminal work of Roundy (1985, 1986) and Jackson et al. (1985) remains unsurpassed in efficiently developing provably-good dynamic policies in this context. The principal contribution of this paper consists in developing a wide range of algorithmic ideas and analytical insights around the continuous-time joint replenishment problem, culminating in a deterministic framework for efficiently approximating optimal dynamic policies to any desired level of accuracy. These advances enable us to derive a compactly-encoded replenishment policy whose long-run average cost is within factor $1 + \epsilon$ of the dynamic optimum, arriving at an efficient polynomial-time approximation scheme (EPTAS). Technically speaking, our approach hinges on affirmative resolutions to two fundamental open questions: -- We devise the first efficient discretization-based framework for approximating the joint replenishment problem. Specifically, we prove that every continuous-time infinite-horizon instance can be reduced to a corresponding discrete-time $O( \frac{ n^3 }{ \epsilon^6 } )$-period instance, while incurring a multiplicative optimality loss of at most $1 + \epsilon$. -- Motivated by this relation, we substantially improve on the $O( 2^{2^{O(1/\epsilon)}} \cdot (nT)^{ O(1) } )$-time approximation scheme of Nonner and Sviridenko (2013) for the discrete-time joint replenishment problem. Beyond an exponential improvement in running time, we demonstrate that randomization and hierarchical decompositions can be entirely avoided, while concurrently offering a relatively simple analysis.

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arXiv2025

New Approximation Guarantees for The Inventory Staggering Problem

Since its inception in the mid-60s, the inventory staggering problem has been explored and exploited in a wide range of application domains, such as production planning, stock control systems, warehousing, and aerospace/defense logistics. However, even with a rich history of academic focus, we are still very much in the dark when it comes to cornerstone computational questions around inventory staggering and to related structural characterizations, with our methodological toolbox being severely under-stocked. The central contribution of this paper consists in devising a host of algorithmic techniques and analytical ideas -- some being entirely novel and some leveraging well-studied concepts in combinatorics and number theory -- for surpassing essentially all known approximation guarantees for the inventory staggering problem. In particular, our work demonstrates that numerous structural properties open the door for designing polynomial-time approximation schemes, including polynomially-bounded cycle lengths, constantly-many distinct time intervals, so-called nested instances, and pairwise coprime settings. These findings offer substantial improvements over currently available constant-factor approximations and resolve outstanding open questions in their respective contexts. In parallel, we develop new theory around a number of yet-uncharted questions, related to the sampling complexity of peak inventory estimation as well as to the plausibility of groupwise synchronization. Interestingly, we establish the global nature of inventory staggering, proving that there are $n$-item instances where, for every subset of roughly $\sqrt{n}$ items, no policy improves on the worst-possible one by a factor greater than $1+\epsilon$, whereas for the entire instance, there exists a policy that outperforms the worst-possible one by a factor of nearly $2$, which is optimal.

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arXiv2024

New Approximation Guarantees for The Economic Warehouse Lot Scheduling Problem

In this paper, we present long-awaited algorithmic advances toward the efficient construction of near-optimal replenishment policies for a true inventory management classic, the economic warehouse lot scheduling problem. While this paradigm has accumulated a massive body of surrounding literature since its inception in the late '50s, we are still very much in the dark as far as basic computational questions are concerned, perhaps due to the evasive nature of dynamic policies in this context. The latter feature forced earlier attempts to either study highly-structured classes of policies or to forgo provably-good performance guarantees altogether; to this day, rigorously analyzable results have been few and far between. The current paper develops novel analytical foundations for directly competing against dynamic policies. Combined with further algorithmic progress and newly-gained insights, these ideas culminate to a polynomial-time approximation scheme for constantly-many commodities as well as to a proof-of-concept $(2-\frac{17}{5000} + \epsilon)$-approximation for general problem instances. In this regard, the efficient design of $\epsilon$-optimal dynamic policies appeared to have been out of reach, since beyond algorithmic challenges by themselves, even the polynomial-space representation of such policies has been a fundamental open question. On the other front, our sub-$2$-approximation constitutes the first improvement over the performance guarantees achievable via ``stationary order sizes and stationary intervals'' (SOSI) policies, which have been state-of-the-art since the mid-'90s.

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arXiv2024

Revenue Maximization in Choice-Based Matching Markets

The primary contribution of this paper resides in devising constant-factor approximation guarantees for revenue maximization in two-sided matching markets, under general pairwise rewards. A major distinction between our work and state-of-the-art results in this context (Ashlagi et al., 2022; Torrico et al., 2023) is that, for the first time, we are able to address reward maximization, reflected by assigning each customer-supplier pair an arbitrarily-valued reward. The specific type of performance guarantees we attain depends on whether one considers the customized model or the inclusive model. The fundamental difference between these settings lies in whether the platform should display to each supplier all selecting customers, as in the inclusive model, or whether the platform can further personalize this set, as in the customized model. Technically speaking, our algorithmic approach and its analysis revolve around presenting novel linear relaxations, leveraging convex stochastic orders, employing approximate dynamic programming, and developing tailor-made analytical ideas. In both models considered, these ingredients allow us to overcome the lack of submodularity and subadditivity that stems from pairwise rewards, plaguing the applicability of existing methods.

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arXiv2024

Improved Approximation Guarantees for Joint Replenishment in Continuous Time

The primary objective of this work is to revisit and revitalize one of the most fundamental models in deterministic inventory management, the continuous-time joint replenishment problem. Our main contribution consists of resolving several long-standing open questions in this context. For most of these questions, we obtain the first quantitative improvement over power-of-$2$ policies and their nearby derivatives, which have been state-of-the-art in terms of provable performance guarantees since the mid-80's.

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arXiv2023

Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps

Motivated by modern-day applications such as Attended Home Delivery and Preference-based Group Scheduling, where decision makers wish to steer a large number of customers toward choosing the exact same alternative, we introduce a novel class of assortment optimization problems, referred to as Maximum Load Assortment Optimization. In such settings, given a universe of substitutable products, we are facing a stream of customers, each choosing between either selecting a product out of an offered assortment or opting to leave without making a selection. Assuming that these decisions are governed by the Multinomial Logit choice model, we define the random load of any underlying product as the total number of customers who select it. Our objective is to offer an assortment of products to each customer so that the expected maximum load across all products is maximized. We consider both static and dynamic formulations. In the static setting, a single offer set is carried throughout the entire process of customer arrivals, whereas in the dynamic setting, the decision maker offers a personalized assortment to each customer, based on the entire information available at that time. The main contribution of this paper resides in proposing efficient algorithmic approaches for computing near-optimal static and dynamic assortment policies. In particular, we develop a polynomial-time approximation scheme (PTAS) for the static formulation. Additionally, we demonstrate that an elegant policy utilizing weight-ordered assortments yields a 1/2- approximation. Concurrently, we prove that such policies are sufficiently strong to provide a 1/4-approximation with respect to the dynamic formulation, establishing a constant-factor bound on its adaptivity gap. Finally, we design an adaptive policy whose expected maximum load is within factor 1-\eps of optimal, admitting a quasi-polynomial time implementation.

math.OC↗

arXiv2023

Assortment Optimization with Visibility Constraints

Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, that we refer to as APV. Here, we are given a universe of substitutable products and a stream of customers. The objective is to determine the optimal assortment of products to offer to each customer in order to maximize the total expected revenue, subject to exogenously-given visibility constraints, stating that each product should be shown to a minimum number of customers. We assume that customer choices follow a Multinomial Logit model (MNL). We provide a structural characterization of optimal assortments and present a linear time algorithm for solving APV. To this end, we introduce a novel function called the ``expanded revenue" of an assortment and establish its supermodularity; our algorithm takes advantage of this structural property. Additionally, we prove that APV can be formulated as a compact linear program. Next, we consider APV with cardinality constraints, which limit the maximum number of products that can be included in an assortment. We prove this problem to be strongly NP-hard and not admitting a Fully Polynomial Time Approximation Scheme (FPTAS), even when all products have identical prices. Subsequently, we devise a Polynomial Time Approximation Scheme (PTAS) for APV under cardinality constraints with identical prices. We also examine the revenue loss resulting from the enforcement of visibility constraints, comparing it to the unconstrained problem. To offset this loss, we propose a novel strategy to distribute the loss incurred among the products subject to visibility constraints, charging each vendor an amount proportional to their product's contribution to the revenue loss.

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arXiv2023

The Continuous-Time Joint Replenishment Problem: $\epsilon$-Optimal Policies via Pairwise Alignment

The main contribution of this paper resides in developing a new algorithmic approach for addressing the continuous-time joint replenishment problem, termed $\Psi$-pairwise alignment. The latter mechanism, through which we synchronize multiple Economic Order Quantity models, allows us to devise a purely-combinatorial algorithm for efficiently approximating optimal policies within any degree of accuracy. As a result, our work constitutes the first quantitative improvement over power-of-$2$ policies, which have been state-of-the-art in this context since the mid-80's. Moreover, in light of recent intractability results, by proposing an efficient polynomial-time approximation scheme (EPTAS) for the joint replenishment problem, we resolve the long-standing open question regarding the computational complexity of this classical setting.

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arXiv2022

A Constructive Prophet Inequality Approach to The Adaptive ProbeMax Problem

In the adaptive ProbeMax problem, given a collection of mutually-independent random variables $X_1, \ldots, X_n$, our goal is to design an adaptive probing policy for sequentially sampling at most $k$ of these variables, with the objective of maximizing the expected maximum value sampled. In spite of its stylized formulation, this setting captures numerous technical hurdles inherent to stochastic optimization, related to both information structure and efficient computation. For these reasons, adaptive ProbeMax has served as a test bed for a multitude of algorithmic methods, and concurrently as a popular teaching tool in courses and tutorials dedicated to recent trends in optimization under uncertainty. The main contribution of this paper consists in proposing a novel method for upper-bounding the expected maximum reward of optimal adaptive probing policies, based on a simple min-max problem. Equipped with this method, we devise purely-combinatorial algorithms for deterministically computing feasible sets whose vicinity to the adaptive optimum is analyzed through prophet inequality ideas. Consequently, this approach allows us to establish improved constructive adaptivity gaps for the ProbeMax problem in its broadest form, where $X_1, \ldots, X_n$ are general random variables, making further advancements when $X_1, \ldots, X_n$ are continuous.

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arXiv2022

Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers

We consider a multi-stage stochastic optimization problem originally introduced by Cygan et al. (2013), studying how a single server should prioritize stochastically departing customers. In this setting, our objective is to determine an adaptive service policy that maximizes the expected total reward collected along a discrete planning horizon, in the presence of customers who are independently departing between one stage and the next with known stationary probabilities. In spite of its deceiving structural simplicity, we are unaware of non-trivial results regarding the rigorous design of optimal or truly near-optimal policies at present time. Our main contribution resides in proposing a quasi-polynomial-time approximation scheme for adaptively serving impatient customers. Specifically, letting $n$ be the number of underlying customers, our algorithm identifies in $O( n^{ O_{ \epsilon }( \log^2 n ) } )$ time an adaptive service policy whose expected reward is within factor $1 - \epsilon$ of the optimal adaptive reward. Our method for deriving this approximation scheme synthesizes various stochastic analyses in order to investigate how the adaptive optimum is affected by alteration to several instance parameters, including the reward values, the departure probabilities, and the collection of customers itself.

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arXiv2020

An Approximate Dynamic Programming Approach to The Incremental Knapsack Problem

We study the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. While various approximation algorithms were developed under mitigating structural assumptions, obtaining non-trivial performance guarantees for this problem in its utmost generality has remained an open question thus far. In this paper, we devise a polynomial-time approximation scheme for general instances of the incremental knapsack problem, which is the strongest guarantee possible given existing hardness results. In contrast to earlier work, our algorithmic approach exploits an approximate dynamic programming formulation. Starting with a simple exponentially sized dynamic program, we prove that an appropriate composition of state pruning ideas yields a polynomially sized state space with negligible loss of optimality. The analysis of this formulation synthesizes various techniques, including new problem decompositions, parsimonious counting arguments, and efficient rounding methods, that may be of broader interest.

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arXiv2020

Approximation Algorithms for The Generalized Incremental Knapsack Problem

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time periods with non-decreasing capacities $W_1 \leq \dots \leq W_T$. When item $i$ is inserted at time $t$, we gain a profit of $p_{it}$; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time $(\frac{1}{2}-ε)$-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

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