arXiv subjects

Rajan Udwani

Publications and source records attributed to Rajan Udwani.

18 recordsLinked to original sources

arXiv2026

Sequential Pricing with Deadlines and Correlated Buyers

We study sequential posted pricing for selling a single item under an exogenous deadline: a seller makes take-it-or-leave-it price offers to buyers one at a time, stopping when some buyer accepts or the selling opportunity expires. We consider both deterministic deadlines and random deadlines drawn from a known distribution, and allow buyer valuations to be either independent or arbitrarily correlated. Despite its practical relevance, this collection of deadline-constrained sequential pricing models has received limited algorithmic study beyond special cases. We develop the first approximation algorithms for revenue maximization across these settings.

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arXiv2026

Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation

Linear-programming (LP)-based primal-dual methods are fundamental for designing and analyzing algorithms in adversarial (prior-free) online resource allocation. This chapter provides a tutorial on two modern primal-dual frameworks, emphasizing recent developments and contemporary models in operations research. Part~I develops an LP-based convex-programming framework where solving a regularized convex program at each arrival captures the tradeoff between greediness and hedging, yielding a dual certificate via Karush-Kuhn-Tucker (KKT) conditions. Because standard LP relaxations can be weak or intractable for stochastic outcomes, Part~II introduces a complementary LP-free framework that provides a universal certificate system for evaluating competitive ratios under such uncertainty. Covering a wide array of models -- including online vertex-weighted bipartite matching, edge-weighted online matching with free disposal, online matching with stochastic rewards, reusable resources, two-sided assortment optimization, configuration allocation (whole-page optimization), AdWords, and costly cancellations -- the tutorial equips readers with versatile proof templates to analyze existing algorithms and develop new solutions for emerging applications.

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arXiv2026

Optimality of Non-Adaptive Algorithms in Online Submodular Welfare Maximization with Stochastic Outcomes

We generalize the problem of online submodular welfare maximization to incorporate various stochastic elements that have gained significant attention in recent years. We show that a non-adaptive Greedy algorithm, which is oblivious to the realization of these stochastic elements, achieves the best possible competitive ratio among all polynomial-time algorithms, including adaptive ones, unless NP$=$RP. This result holds even when the objective function is not submodular but instead satisfies the weaker submodular order property. Our results unify and strengthen existing competitive ratio bounds across well-studied settings and diverse arrival models, showing that, in general, adaptivity to stochastic elements offers no advantage in terms of competitive ratio. To establish these results, we introduce a technique that lifts known results from the deterministic setting to the generalized stochastic setting. The technique has broad applicability, enabling us to show that, in certain special cases, non-adaptive Greedy-like algorithms outperform the Greedy algorithm and achieve the optimal competitive ratio. We also apply the technique in reverse to derive new upper bounds on the performance of Greedy-like algorithms in deterministic settings by leveraging upper bounds on the performance of non-adaptive algorithms in stochastic settings.

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arXiv2025

When Location Shapes Choice: Placement Optimization of Substitutable Products

Strategic product placement can have a strong influence on customer purchase behavior in physical stores as well as online platforms. Motivated by this, we consider the problem of optimizing the placement of substitutable products in designated display locations to maximize the expected revenue of the seller. We model the customer behavior as a two-stage process: first, the customer visits a subset of display locations according to a browsing distribution; second, the customer chooses at most one product from the displayed products at those locations according to a choice model. Our goal is to design a general algorithm that can select and place the products optimally for any browsing distribution and choice model, and we call this the Placement problem. We give a randomized algorithm that utilizes an $α$-approximate algorithm for cardinality constrained assortment optimization and outputs a $\frac{Θ(α)}{\log m}$-approximate solution (in expectation) for Placement with $m$ display locations, i.e., our algorithm outputs a solution with value at least $\frac{Ω(α)}{\log m}$ factor of the optimal and this is tight in the worst case. We also give algorithms with stronger guarantees in some special cases. In particular, we give a deterministic $\frac{Ω(1)}{\log m}$-approximation algorithm for the Markov choice model, and a tight $(1-1/e)$-approximation algorithm for the problem when products have identical prices.

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arXiv2025

A Markovian Approach for Cross-Category Complementarity in Choice Modeling

While single-purchase choice models have been widely studied in assortment optimization, customers in modern retail and e-commerce environments often purchase multiple items across distinct product categories, exhibiting both substitution and complementarity. We consider the cross-category assortment optimization problem where retailers jointly determine assortments across categories to maximize expected revenue. Most prior work on the topic either overlooks complementarity or proposes models that lead to intractable optimization problems, despite being based on the multinomial logit (MNL) choice model. We propose a sequential multi-purchase choice model for cross-category choice that incorporates complementarity through a Markovian transition structure across categories, while allowing general Random Utility Maximization (RUM)-based choice models to capture the within-category substitution. We develop an Expectation-Maximization algorithm for estimation, and a polynomial-time algorithm for unconstrained assortment optimization that yields the optimal solution when the within-category substitution follows a Markov chain choice model. Furthermore, we introduce an empirical metric to quantify complementarity strength across product categories and conduct extensive numerical experiments on both synthetic data and a large-scale transaction-level dataset from a major US grocery store. Our model yields improvements in predictive accuracy, model fit, and expected revenue in setting with complementarity, and it reveals intuitive market structures such as brand-loyal cross-category purchasing. Overall, we believe that our model provides a theoretically-grounded and practical framework for modeling complementarity and making better cross-category assortment decisions.

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arXiv2025

A Black-Box Approach for Exogenous Replenishment in Online Resource Allocation

In a typical online resource allocation problem, we start with a fixed inventory of resources and make online allocation decisions in response to resource requests that arrive sequentially over a finite horizon. We consider settings where the inventory is replenished over time according to an unknown exogenous process. We introduce black-box methods that extend any existing algorithm, originally designed without considering replenishment, into one that works with an arbitrary (adversarial or stochastic) replenishment process. Our approach preserves the original algorithm's competitive ratio in regimes with large initial inventory, thereby enabling the seamless integration of exogenous replenishment into a large body of existing algorithmic results for both adversarial and stochastic arrival models.

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arXiv2025

A Unified Algorithmic Framework for Dynamic Assortment Optimization under MNL Choice

We consider assortment and inventory planning problems with dynamic stockout-based substitution effects, and without replenishment, in two different settings: (1) Customers can see all available products when they arrive, a typical scenario in physical stores. (2) The seller can choose to offer a subset of available products to each customer, which is more common on online platforms. Both settings are known to be computationally challenging, and the current approximation algorithms for the two settings are quite different. We develop a unified algorithm framework under the MNL choice model for both settings. Our algorithms improve on the state-of-the-art algorithms in terms of approximation guarantee and runtime, and the ability to manage uncertainty in the total number of customers and handle more complex constraints. In the process, we establish various novel properties of dynamic assortment planning (for the MNL choice model) that may be useful more broadly.

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arXiv2024

Asymptotically Optimal Competitive Ratio for Online Allocation of Reusable Resources

We consider the problem of online allocation (matching, budgeted allocations, and assortments) of reusable resources where an adversarial sequence of resource requests is revealed over time and any allocated resource is used/rented for a stochastic duration drawn independently from a resource dependent usage distribution. Previously, it was known that a greedy algorithm is 0.5--competitive against the clairvoyant benchmark that knows the entire sequence of requests in advance (Gong et al. (2021)). We give a novel algorithm that is $(1-1/e)$--competitive for arbitrary usage distributions when the starting capacity of each resource is large and the usage distributions are known. This is the best achievable competitive ratio guarantee for the problem, i.e., no online algorithm can have better competitive ratio. We also give a distribution oblivious online algorithm and show that it is $(1-1/e)$--competitive in special cases. At the heart of our algorithms is a new quantity that factors in the potential of reusability for each resource by (computationally) creating an asymmetry between identical units of the resource. We establish the performance guarantee for our algorithms by constructing a feasible solution to a novel system of inequalities that allows direct comparison with the clairvoyant benchmark instead of a linear programming (LP) relaxation of the benchmark. Our technique generalizes the primal-dual analysis framework for online resource allocation and may be of broader interest.

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arXiv2024

Adwords with Unknown Budgets and Beyond

In the classic Adwords problem introduced by Mehta et al.\ (2007), we have a bipartite graph between advertisers and queries. Each advertiser has a maximum budget that is known a priori. Queries are unknown a priori and arrive sequentially. When a query arrives, advertisers make bids and we (immediately and irrevocably) decide which (if any) Ad to display based on the bids and advertiser budgets. The winning advertiser for each query pays their bid up to their remaining budget. Our goal is to maximize total budget utilized without any foreknowledge of the arrival sequence (which could be adversarial). We consider the setting where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded. A naïve greedy algorithm is 0.5 competitive for this setting and finding an algorithm with better performance remained an open problem. We show that no deterministic algorithm has competitive ratio better than 0.5 and give the first (randomized) algorithm with strictly better performance guarantee. We show that the competitive ratio of our algorithm is at least 0.522 but also strictly less than $(1-1/e)$. We present novel applications of budget oblivious algorithms in search ads and beyond. In particular, we show that our algorithm achieves the best possible performance guarantee for deterministic online matching in the presence of multi-channel traffic (Manshadi et al. (2022)).

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arXiv2024

Submodular Order Functions and Assortment Optimization

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. We give fast algorithms with strong approximation guarantees for maximizing submodular order functions under a variety of constraints and show a nearly tight upper bound on the highest approximation guarantee achievable by algorithms with polynomial query complexity. Applying this new notion to the problem of constrained assortment optimization in fundamental choice models, we obtain new algorithms that are both faster and have stronger approximation guarantees (in some cases, first algorithm with constant factor guarantee). We also show an intriguing connection to the maximization of monotone submodular functions in the streaming model, where we recover best known approximation guarantees as a corollary of our results.

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arXiv2024

Almost Tight Bounds for Online Hypergraph Matching

In the online hypergraph matching problem, hyperedges of size $k$ over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A naïve greedy algorithm for this problem achieves a competitive ratio of $\frac{1}{k}$. We show that no (randomized) online algorithm has competitive ratio better than $\frac{2+o(1)}{k}$. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio $\frac{1-o(1)}{\ln(k)}$ and show that no online algorithm can have competitive ratio strictly better than $\frac{1+o(1)}{\ln(k)}$. Lastly, we give a $\frac{1-o(1)}{\ln(k)}$ competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.

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arXiv2024

When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching

We study the problem of vertex-weighted online bipartite matching with stochastic rewards where matches may fail with some known probability and the decision maker has to adapt to the sequential realization of these outcomes. Recent works have studied several special cases of this problem and it was known that the (randomized) Perturbed Greedy algorithm due to Aggarwal et al. (SODA, 2011) achieves the best possible competitive ratio guarantee of $(1-1/e)$ in some cases. We give a simple proof of these results by reducing (special cases of) the stochastic rewards problem to the deterministic setting of online bipartite matching (Karp, Vazirani, Vazirani (STOC, 1990)). More broadly, our approach gives conditions under which it suffices to analyze the competitive ratio of an algorithm for the simpler setting of deterministic rewards in order to obtain a competitive ratio guarantee for stochastic rewards. The simplicity of our approach reveals that the Perturbed Greedy algorithm has a competitive ratio of $(1-1/e)$ even in certain settings with correlated rewards, where no results were previously known. Finally, we show that without any special assumptions, the Perturbed Greedy algorithm has a competitive ratio strictly less than $(1-1/e)$ for vertex-weighted online matching with stochastic rewards.

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arXiv2022

Online Bipartite Matching with Reusable Resources

We study the classic online bipartite matching problem with a twist: offline vertices, called resources, are $\textit{reusable}$. In particular, when a resource is matched to an online vertex it is unavailable for a deterministic time duration $d$ after which it becomes available again for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem have resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of $\textit{unit inventory}$ and give the first algorithms that are provably better than the naive greedy approach which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of $0.589$, generalizes the classic RANKING algorithm for online bipartite matching of non-reusable resources (Karp et al., 1990), by $\textit{reranking}$ resources independently over time. While reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of $0.505$, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex, and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al., 2020) to reusable resources, in order to induce negative correlation in our tie breaking step and to beat the competitive ratio of $0.5$. Both of our results also extend to the case where offline vertices have weights.

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arXiv2022

Periodic Reranking for Online Matching of Reusable Resources

We consider a generalization of the vertex weighted online bipartite matching problem where the offline vertices, called resources, are reusable. In particular, when a resource is matched it is unavailable for a deterministic time duration $d$ after which it becomes available for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem has resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of unit inventory and give the first algorithm that is provably better than the naïve greedy approach which has a competitive ratio of (exactly) 0.5. In particular, we achieve a competitive ratio of 0.589 against an LP relaxation of the offline problem.

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arXiv2022

Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of $0.567$ for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being $1/2$ were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is $(1-1/e)$ competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic $0.596$ competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based analysis. This allows us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches.

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arXiv2021

Multi-Objective Maximization of Monotone Submodular Functions with Cardinality Constraint

We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. While it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, Krause et al.\ (2008) showed that when the number of objectives $m$ grows as the cardinality $k$ i.e., $m=Ω(k)$, the problem is inapproximable (unless $P=NP$). On the other hand, when $m$ is constant Chekuri et al.\ (2010) showed a randomized $(1-1/e)-ε$ approximation with runtime (number of queries to function oracle) $n^{m/ε^3}$. %In fact, the result of Chekuri et al.\ (2010) is for the far more general case of matroid constant. We focus on finding a fast and practical algorithm that has (asymptotic) approximation guarantees even when $m$ is super constant. We first modify the algorithm of Chekuri et al.\ (2010) to achieve a $(1-1/e)$ approximation for $m=o(\frac{k}{\log^3 k})$. This demonstrates a steep transition from constant factor approximability to inapproximability around $m=Ω(k)$. Then using Multiplicative-Weight-Updates (MWU), we find a much faster $\tilde{O}(n/δ^3)$ time asymptotic $(1-1/e)^2-δ$ approximation. While the above results are all randomized, we also give a simple deterministic $(1-1/e)-ε$ approximation with runtime $kn^{m/ε^4}$. Finally, we run synthetic experiments using Kronecker graphs and find that our MWU inspired heuristic outperforms existing heuristics.

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arXiv2020

Online Allocation of Reusable Resources via Algorithms Guided by Fluid Approximations

We consider the problem of online allocation (matching and assortments) of reusable resources where customers arrive sequentially in an adversarial fashion and allocated resources are used or rented for a stochastic duration that is drawn independently from known distributions. Focusing on the case of large inventory, we give an algorithm that is $(1-1/e)$ competitive for general usage distributions. At the heart of our result is the notion of a relaxed online algorithm that is only subjected to fluid approximations of the stochastic elements in the problem. The output of this algorithm serves as a guide for the final algorithm. This leads to a principled approach for seamlessly addressing stochastic elements (such as reusability, customer choice, and combinations thereof) in online resource allocation problems, that may be useful more broadly.

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arXiv2017

Robust Monotone Submodular Function Maximization

We consider a robust formulation, introduced by Krause et al. (2008), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to $τ$ elements from the chosen set. For the fundamental case of $τ=1$, we give a deterministic $(1-1/e)-1/Θ(m)$ approximation algorithm, where $m$ is an input parameter and number of queries scale as $O(n^{m+1})$. In the process, we develop a deterministic $(1-1/e)-1/Θ(m)$ approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (2010), we show a randomized $(1-1/e)-ε$ approximation for constant $τ$ and $ε\leq \frac{1}{\tildeΩ(τ)}$, making $O(n^{1/ε^3})$ queries. Further, for $τ\ll \sqrt{k}$, we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.

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