arXiv subjects

Rad Niazadeh

Publications and source records attributed to Rad Niazadeh.

At least 19 recordsLinked to original sources

arXiv2026

Stationary Online Contention Resolution Schemes

Online contention resolution schemes (OCRSs) are a central tool in Bayesian online selection and resource allocation: they convert fractional ex-ante relaxations into feasible online policies while preserving each marginal probability up to a constant factor. Despite their importance, designing (near) optimal OCRSs is often technically challenging, and many existing constructions rely on indirect reductions to prophet inequalities and LP duality, resulting in algorithms that are difficult to interpret or implement. In this paper, we introduce "stationary online contention resolution schemes (S-OCRSs)," a permutation-invariant class of OCRSs in which the distribution of the selected feasible set is independent of arrival order. We show that S-OCRSs admit an exact distributional characterization together with a universal online implementation. We then develop a general `maximum-entropy' approach to construct and analyze S-OCRSs, reducing the design of online policies to constructing suitable distributions over feasible sets. This yields a new technical framework for designing simple and possibly improved OCRSs. We demonstrate the power of this framework across several canonical feasibility environments. In particular, we obtain an improved $(3-\sqrt{5})/2$-selectable OCRS for bipartite matchings, attaining the independence benchmark conjectured to be optimal and yielding the best known prophet inequality for this setting. We also obtain a $1-\sqrt{2/(πk)} + O(1/k)$-selectable OCRS for $k$-uniform matroids and a simple, explicit $1/2$-selectable OCRS for weakly Rayleigh matroids (including all $\mathbb{C}$-representable matroids such as graphic and laminar). While these guarantees match the best known bounds, our framework also yields concrete and systematic constructions, providing transparent algorithms in settings where previous OCRSs were implicit or technically involved.

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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.

math.OC↗

arXiv2026

Non-Exclusive Notifications for Ride-Hailing at Lyft I: Single-Cycle Approximation Algorithms

Ride-hailing platforms increasingly rely on non-exclusive notifications-broadcasting a single request to multiple drivers simultaneously-to mitigate inefficiencies caused by uncertain driver acceptance. In this paper, the first in a two-part collaboration with Lyft, we formally model the 'Notification Set Selection Problem' for a single decision cycle, where the platform determines the optimal subset of drivers to notify for each incoming ride request. We analyze this combinatorial optimization problem under two contention-resolution protocols: 'First Acceptance (FA)', which prioritizes speed by assigning the ride to the first responder, and 'Best Acceptance (BA)', which prioritizes match quality by selecting the highest-valued accepting driver. We show that welfare maximization under both mechanisms is strongly NP-hard, ruling out a Fully Polynomial Time Approximation Scheme (FPTAS). Despite this, we derive several positive algorithmic results. For FA, we present a Polynomial Time Approximation Scheme (PTAS) for the single-rider case and a constant-factor approximation (factor 4) for the general matching setting. We highlight that the FA valuation function can be viewed as a novel discrete choice model with theoretical properties of independent interest. For BA, we prove that the objective is monotone and submodular, admitting a standard $(1 - 1/e)$-approximation. Moreover, using a polynomial-time demand oracle that we design for this problem, we show it is possible to surpass the $(1 - 1/e)$ barrier. Finally, in the special case of homogeneous acceptance probabilities, we show that the BA problem can be solved exactly in polynomial time via a linear programming formulation. We validate the empirical performance our algorithms through numerical experiments on synthetic data and on instances calibrated using real ride-sharing data from Lyft.

cs.DS↗

arXiv2026

Non-Exclusive Notifications for Ride-Hailing at Lyft II: Simulations and Marketplace Analysis

Ride-hailing platforms increasingly face uncertain driver acceptance, which makes traditional one-to-one 'exclusive dispatch (ED)' less efficient: rejections and timeouts force sequential retries and lengthen rider wait times, which in turn creates friction in the marketplace. 'Non-exclusive dispatch (NED)' mitigates this friction by broadcasting a request to multiple drivers in parallel. While NED can reduce latency, it introduces new design challenges -- most notably, how to choose notification sets and how to resolve driver contention (when multiple drivers accept the same ride). In this paper -- the second in a two-part collaboration with Lyft -- we develop a theoretically grounded framework to evaluate the long-run performance and marketplace effects of transitioning from ED to NED. We bridge theory and practice by combining (i) an optimization model that formulates NED as a constrained welfare maximization problem with (ii) large-scale discrete-event simulations on proprietary Lyft traces and (iii) a stylized macroscopic equilibrium model. Across simulation and equilibrium analysis, we find that NED improves key fulfillment metrics relative to ED: it reduces match time (and hence rider reneging) while increasing both the number and the average quality of completed matches. We also quantify the speed--quality trade-off between two common contention resolution rules, 'First-Accept' and 'Best-Accept': First-Accept maximizes speed and throughput, whereas Best-Accept is required to maximize per-match quality. Finally, we show that slightly conservative notification heuristics can improve long-run efficiency by avoiding excessive locking of high-value drivers and preserving future availability.

cs.GT↗

arXiv2025

Near-Optimal Bayesian Online Assortment of Reusable Resources

Motivated by the applications of rental services in e-commerce, we consider revenue maximization in online assortment of reusable resources for a stream of arriving consumers with different types. We design competitive online algorithms with respect to the optimum online policy in the Bayesian setting, in which types are drawn independently from known heterogeneous distributions over time. In the regime where the minimum of initial inventories $c_0$ is large, our main result is a near-optimal $1-\min\left(\frac{1}{2},\sqrt{\log(c_0)/c_0}\right)$ competitive algorithm for the general case of reusable resources. Our algorithm relies on an expected LP benchmark for the problem, solves this LP, and simulates the solution through an independent randomized rounding. The main challenge is obtaining point-wise inventory feasibility in a computationally efficient fashion from these simulation-based algorithms. To this end, we use several technical ingredients to design $\textit{discarding policies}$ -- one for each resource. These policies handle the trade-off between the inventory feasibility under reusability and the revenue loss of each of the resources. However, discarding a unit of a resource changes the future consumption of other resources. To handle this new challenge, we also introduce $\textit{post-processing}$ assortment procedures that help with designing and analyzing our discarding policies as they run in parallel, which might be of independent interest. As a side result, by leveraging techniques from the literature on prophet inequality, we further show an improved near-optimal $1-1/\sqrt{c_0+3}$ competitive algorithm for the special case of non-reusable resources. We finally evaluate the performance of our algorithms using the numerical simulations on the synthetic data.

cs.DS↗

arXiv2025

Robustness of Online Inventory Balancing to Inventory Shocks

In classic adversarial online resource allocation problems such as AdWords, customers arrive online while products are given offline with a fixed initial inventory. To ensure revenue guarantees under uncertainty, the decision maker must balance consumption across products. Based on this, the prevalent policy "inventory balancing (IB)" has proved to be optimal or near-optimal competitive in almost all classic settings. However, these models do not capture various forms of inventory shocks on the supply side, which play an important role in real-world online assortment and can significantly impact the revenue performance of the IB algorithm. Motivated by this paradigm, we introduce a variant of online assortment planning with inventory shocks. Our model considers adversarial exogenous shocks (where supply increases unpredictably) and allocation-coupled endogenous shocks (where an inventory reduction is triggered by the algorithms and re-adjusted after a usage duration), whose combination leads to non-monotonic inventory fluctuations. As our main result, we show the robustness of IB-type strategies against such shocks by designing a new family of optimal competitive algorithms called "Batched Inventory Balancing (BIB)." Using a novel randomized primal-dual method, we bound the competitive ratio of BIB against optimal offline. We show that with proper choice of a certain parameter, this competitive ratio is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model. Moreover, we characterize BIB's competitive ratio parametric by its penalty function and show that it matches exactly the competitive ratio of IB without shocks. Our refined analysis reduces the dual construction to a combinatorial "interval assignment problem" whose algorithmic solution may be of independent interest.

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arXiv2025

Markovian Search with Ex-Ante Constraints: Theory and Applications to Socially Aware Algorithmic Hiring

We develop an algorithmic framework to incorporate "ex-ante" constraints on outcomes (that hold only on average) into stateful sequential search with costly inspection. Our framework encompasses the classical Weitzman's Pandora's box [Weitzman, 1979] as well as its extensions to joint Markovian scheduling [Dumitriu et al., 2003; Gittins, 1979], modeling richer processes such as multistage search with multiple layers of inspection. Ex-ante constraints in search are particularly motivated by social considerations in algorithmic hiring, where they adjust outcome distributions to promote equity and access. Building on the optimality of index-based policies in the unconstrained problems, we show that optimal policies under a single ex-ante constraint (e.g., demographic parity) retain an index-based structure but require (i) dual-based adjustments of the indices and (ii) randomization between two such adjustments via a "tie-breaking rule," both easy to compute and economically interpretable. We then extend our results to handle multiple affine constraints by reduction to a variant of the exact Carathéodory problem and providing a polynomial-time algorithm to construct an optimal randomized dual-adjusted index-based policy that satisfies all constraints simultaneously. For general affine and convex constraints, we develop a primal-dual algorithm that randomizes over a polynomial number of dual-based adjustments, yielding a near-feasible, near-optimal policy. All these results rely on the key observation that a suitable relaxation of the Lagrange dual function for these constrained problems admits index-based policies akin to those in the unconstrained setting. Finally, through a numerical study, we investigate the implications of various socially aware ex-ante constraints on the utilitarian loss (price of fairness), and examine whether they achieve their intended socially desirable outcomes.

cs.DS↗

arXiv2025

Robust Dynamic Staffing with Predictions

We consider a natural dynamic staffing problem in which a decision-maker sequentially hires workers over a finite horizon to meet an unknown demand revealed at the end. Predictions about demand arrive over time and become increasingly accurate, while worker availability decreases. This creates a fundamental trade-off between hiring early to avoid understaffing (when workers are more available but forecasts are less reliable) and hiring late to avoid overstaffing (when forecasts are more accurate but availability is lower). This problem is motivated by last-mile delivery operations, where companies such as Amazon rely on gig-economy workers whose availability declines closer to the operating day. To address practical limitations of Bayesian models (in particular, to remain agnostic to the underlying forecasting method), we study this problem under adversarial predictions. In this model, sequential predictions are adversarially chosen uncertainty intervals that (approximately) contain the true demand. The objective is to minimize worst-case staffing imbalance cost. Our main result is a simple and computationally efficient online algorithm that is minimax optimal. We first characterize the minimax cost against a restricted adversary via a polynomial-size linear program, then show how to emulate this solution in the general case. While our base model focuses on a single demand, we extend the framework to multiple demands (with egalitarian/utilitarian objectives), to settings with costly reversals of hiring decisions, and to inconsistent prediction intervals. We also introduce a practical "re-solving" variant of our algorithm, which we prove is also minimax optimal. Finally we conduct numerical experiments showing that our algorithms outperform Bayesian heuristics in both cost and speed, and are competitive with (approximate or exact) Bayesian-optimal policies when those can be computed.

cs.DS↗

arXiv2025

Online Resource Allocation with Cancellations

We initiate the study of two-sided online resource allocation with costly cancellations. Our focus is on edge-weighted online bipartite matching (and several of its extensions), where nodes arrive online and request offline resources. In contrast to the classic literature, any fraction of an offline resource that was preallocated to an earlier online node can be reclaimed, resulting in the loss of the previously allocated edge-weight plus an additional penalty equal to a non-negative constant factor $f$ times the edge-weight. Parameterizing the problem by the buyback factor $f$, our main result is the development of optimal competitive algorithms for \emph{all possible values} of $f$ through a novel primal-dual family of algorithms in the fractional (or equivalently, large capacity) setting, and establishing their optimality by deriving matching lower bounds. Interestingly, our results reveal a phase transition: for the small buyback regime ($f < \frac{e-2}{2}$), the optimal competitive ratio is $\frac{e}{e-(1+f)}$, and for the large buyback regime ($f \geq \frac{e-2}{2}$), the competitive ratio is $-W_{-1}\left(\frac{-1}{e(1+f)}\right)$, where $W_{-1}$ is the non-principal branch of the Lambert $W$ function. We also study variants of this model, such as matching with deterministic integral allocations. We again show a phase transition: for the small buyback regime ($f < \frac{1}{3}$), the optimal competitive ratio is $\frac{2}{1-f}$, while for the large buyback regime ($f \geq \frac{1}{3}$), the competitive ratio is $1 + 2f + 2\sqrt{f(1+f)}$. We further consider various extensions, including to configuration allocations and submodular welfare maximization, as well as negative values of $f$, modeling a secondary supply channels or overflow capacities available at discounted rates. Our unifying primal-dual framework achieves the exact optimal competitive ratio across all these variants

cs.DS↗

arXiv2025

Dynamic Matching with Post-allocation Service and its Application to Refugee Resettlement

Motivated by our collaboration with a major refugee resettlement agency in the U.S., we study a dynamic matching problem where each new arrival (a refugee case) must be matched immediately and irrevocably to one of the static resources (a location with a fixed annual quota). In addition to consuming the static resource, each case requires post-allocation services from a server, such as a translator. Given the uncertainty in service time, a server may not be available at a given time, thus we refer to it as a dynamic resource. Upon matching, the case will wait to avail service in a first-come-first-serve manner. Bursty matching to a location may result in undesirable congestion at its corresponding server. Consequently, the central planner (the agency) faces a dynamic matching problem with an objective that combines the matching reward (captured by pair-specific employment outcomes) with the cost for congestion for dynamic resources and over-allocation for the static ones. Motivated by the observed fluctuations in the composition of refugee pools across the years, we aim to design algorithms that do not rely on distributional knowledge. We develop learning-based algorithms that are asymptotically optimal in certain regimes, easy to interpret, and computationally fast. Our design is based on learning the dual variables of the underlying optimization problem; however, the main challenge lies in the time-varying nature of the dual variables associated with dynamic resources. Our theoretical development brings together techniques from Lyapunov analysis, adversarial online learning, and stochastic optimization. On the application side, when tested on real data from our partner agency and incorporating practical considerations, our method outperforms existing ones making it a viable candidate for replacing the current practice upon experimentation.

cs.DS↗

arXiv2025

Online Job Assignment

Motivated primarily by applications in cloud computing, we study a simple, yet powerful, online allocation problem in which jobs of varying durations arrive over continuous time and must be assigned immediately and irrevocably to one of the available offline servers. Each server has a fixed initial capacity, with assigned jobs occupying one unit for their duration and releasing it upon completion. The algorithm earns a reward for each assignment upon completion. We consider a general heterogeneous setting where both the reward and duration of a job depend on the job-server pair. The objective of the online algorithm is to maximize the total collected reward, and remain competitive against an omniscient benchmark that knows all job arrivals in advance. Our main contribution is the design of a new online algorithm, termed Forward-Looking BALANCE (FLB), and using primal-dual framework to establish that it is (asymptotically) optimal-competitive. This meta-algorithm has two main primitives: (i) keeping track of the capacity used for each server at each time and applying a penalty function to this quantity, and (ii) adjusting the reward of assigning a job to a server by subtracting the total penalty of a particularly chosen subset of future times, in contrast to just looking at the current time. The FLB algorithm then assigns the arriving job to the server with the maximum adjusted reward. If R and D are the ratios of maximum over minimum rewards and durations, we show that the FLB algorithm obtains an asymptotic competitive ratio of ln(RD)+3lnln(max(R,D))+O(1). We further show this bound has optimal dependencies on all the parameters. Our main analysis combines a novel dual-fitting technique, which leverages the configuration LP benchmark for this problem, and a novel inductive argument to establish the capacity feasibility of the algorithm, which might be of independent interest.

cs.DS↗

arXiv2025

Prophet Inequalities with Cancellation Costs

Most of the literature on online algorithms in revenue management focuses on settings with irrevocable decisions, where once a decision is made upon the arrival of a new input, it cannot be canceled later. Motivated by modern applications -- such as cloud spot markets, selling banner ads, or online hotel booking -- we introduce and study "prophet inequalities with cancellations" under linear cancellation costs (known as the buyback model). In the classic prophet inequality problem, a sequence of independent random variables $X_1, X_2, \ldots$ with known distributions is revealed one by one, and a decision maker must decide when to stop and accept the current variable in order to maximize the expected value of their choice. In our model, after accepting $X_j$, one may later discard $X_j$ and accept another $X_i$ at a cost of $f \times X_j$, where $f\geq 0$ is a parameter. The goal is to maximize the expected net reward: the value of the final accepted variable minus the total cancellation cost. We aim to design online policies that are competitive against the optimal offline benchmark. Our first main result is an optimal prophet inequality for all parameters $f \ge 0$. We fully characterize the worst-case competitive ratio of the optimal online policy against the optimal offline via the solution to a certain differential equation (for which we provide a constructive solution). Our second main result is to design and analyze a simple and polynomial-time randomized adaptive policy that achieves this optimal competitive ratio. Importantly, our policy is order-agnostic (à la [Samuel-Cahn, 1984]), as it only needs the set of distributions and not their arrival order. These results are obtained by novel techniques related to factor-revealing LPs and generalized flow, reductions to a differential equation, and embedding of problem instances into specific Poisson point processes.

cs.DS↗

arXiv2024

Linear Programming Based Near-Optimal Pricing for Laminar Bayesian Online Selection

The Bayesian online selection problem aims to design a pricing scheme for a sequence of arriving buyers that maximizes the expected social welfare (or revenue) subject to different structural constraints. Inspired by applications with a hierarchy of service, this paper focuses on the cases where a laminar matroid characterizes the set of served buyers. We give the first Polynomial-Time Approximation Scheme (PTAS) for the problem when the laminar matroid has constant depth. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that approximate the optimum online solution with any degree of accuracy, plus a concentration argument showing that rounding incurs a small loss. We also study another variation, which we call the production-constrained problem. The allowable set of served buyers is characterized by a collection of production and shipping constraints that form a particular example of a laminar matroid. Using a similar LP-based approach, we design a PTAS for this problem, although in this special case the depth of the underlying laminar matroid is not necessarily a constant. The analysis exploits the negative dependency of the optimum selection rule in the lower levels of the laminar family. Finally, to demonstrate the generality of our technique, we employ the linear programming-based approach employed in the paper to re-derive some of the classic prophet inequalities known in the literature -- as a side result.

cs.GT↗

arXiv2024

Misalignment, Learning, and Ranking: Harnessing Users Limited Attention

In digital health and EdTech, recommendation systems face a significant challenge: users often choose impulsively, in ways that conflict with the platform's long-term payoffs. This misalignment makes it difficult to effectively learn to rank items, as it may hinder exploration of items with greater long-term payoffs. Our paper tackles this issue by utilizing users' limited attention spans. We propose a model where a platform presents items with unknown payoffs to the platform in a ranked list to $T$ users over time. Each user selects an item by first considering a prefix window of these ranked items and then picking the highest preferred item in that window (and the platform observes its payoff for this item). We study the design of online bandit algorithms that obtain vanishing regret against hindsight optimal benchmarks. We first consider adversarial window sizes and stochastic iid payoffs. We design an active-elimination-based algorithm that achieves an optimal instance-dependent regret bound of $O(\log(T))$, by showing matching regret upper and lower bounds. The key idea is using the combinatorial structure of the problem to either obtain a large payoff from each item or to explore by getting a sample from that item. This method systematically narrows down the item choices to enhance learning efficiency and payoff. Second, we consider adversarial payoffs and stochastic iid window sizes. We start from the full-information problem of finding the permutation that maximizes the expected payoff. By a novel combinatorial argument, we characterize the polytope of admissible item selection probabilities by a permutation and show it has a polynomial-size representation. Using this representation, we show how standard algorithms for adversarial online linear optimization in the space of admissible probabilities can be used to obtain a polynomial-time algorithm with $O(\sqrt{T})$ regret.

cs.LG↗

arXiv2024

Combinatorial Bernoulli Factories

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has sample access to independent draws of a Bernoulli random variable with mean equal to $p$. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in P$ for a given polytope $P\subset [0,1]^n$, output a randomized vertex such that the expected value of the $i$-th coordinate is \emph{exactly} equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$. We show that a polytope $P$ admits a Bernoulli factory if and and only if $P$ is the intersection of $[0,1]^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $P$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the $k$-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.

cs.DS↗

arXiv2023

Online Learning via Offline Greedy Algorithms: Applications in Market Design and Optimization

Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(\sqrt{T})$ (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a $O(T^{2/3})$ (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We also extend our reduction to greedy-like first order methods used in continuous optimization, such as those used for maximizing continuous strong DR monotone submodular functions subject to convex constraints. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds. We complement our theoretical studies by conducting numerical simulations for two of our applications, in both of which we observe that the numerical performance of our transformations outperforms the theoretical guarantees in practical instances.

cs.LG↗

arXiv2022

Batching and Optimal Multi-stage Bipartite Allocations

In several applications of real-time matching of demand to supply in online marketplaces, the platform allows for some latency to batch the demand and improve the efficiency. Motivated by these applications, we study the optimal trade-off between batching and inefficiency under adversarial arrival. As our base model, we consider K-stage variants of the vertex weighted b-matching in the adversarial setting, where online vertices arrive stage-wise and in K batches -- in contrast to online arrival. Our main result for this problem is an optimal (1-(1-1/K)^K)- competitive (fractional) matching algorithm, improving the classic (1-1/e) competitive ratio bound known for its online variant (Mehta et al., 2007; Aggarwal et al., 2011). We also extend this result to the rich model of multi-stage configuration allocation with free-disposals (Devanur et al., 2016), which is motivated by the display advertising in video streaming platforms. Our main technique is developing tools to vary the trade-off between "greedy-ness" and "hedging" of the algorithm across stages. We rely on a particular family of convex-programming based matchings that distribute the demand in a specifically balanced way among supply in different stages, while carefully modifying the balancedness of the resulting matching across stages. More precisely, we identify a sequence of polynomials with decreasing degrees to be used as strictly concave regularizers of the maximum weight matching linear program to form these convex programs. At each stage, our algorithm returns the corresponding regularized optimal solution as the matching of this stage (by solving the convex program). Using structural properties of these convex programs and recursively connecting the regularizers together, we develop a new multi-stage primal-dual framework to analyze the competitive ratio. We further show this algorithm is optimally competitive.

cs.DS↗

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.

cs.DS↗