arXiv subjects

Pranav Nuti

Publications and source records attributed to Pranav Nuti.

9 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 $1-\sqrt{2/(\pi k)} + O(1/k)$-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.

cs.GT↗

arXiv2024

Static Pricing for Single Sample Multi-unit Prophet Inequalities

In this paper, we study $k$-unit single sample prophet inequalities. A seller has $k$ identical, indivisible items to sell. A sequence of buyers arrive one-by-one, with each buyer's private value for the item, $X_i$, revealed to the seller when they arrive. While the seller is unaware of the distribution from which $X_i$ is drawn, they have access to a single sample, $Y_i$ drawn from the same distribution as $X_i$. What strategies can the seller adopt for selling items so as to maximize social welfare? Previous work has demonstrated that when $k = 1$, if the seller sets a price equal to the maximum of the samples, they can achieve a competitive ratio of $\frac{1}{2}$ of the social welfare, and recently Pashkovich and Sayutina established an analogous result for $k = 2$. In this paper, we prove that for $k \geq 3$, setting a (static) price equal to the $k^{\text{th}}$ largest sample also obtains a competitive ratio of $\frac{1}{2}$, resolving a conjecture Pashkovich and Sayutina pose. We also consider the situation where $k$ is large. We demonstrate that setting a price equal to the $(k-\sqrt{2k\log k})^{\text{th}}$ largest sample obtains a competitive ratio of $1 - \sqrt{\frac{2\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$, and that this is the optimal possible ratio achievable with a static pricing scheme with access to a single sample. This should be compared against a competitive ratio $1 - \sqrt{\frac{\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$, which is the optimal possible ratio achievable with a static pricing scheme with knowledge of the distributions of the values.

cs.GT↗

arXiv2024

Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is $0.382$-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$-selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$-selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order -- give each vertex a random priority and process edges in lexicographic order -- results in a strictly better contention resolution scheme that is $1-\ln(2-1/e)\approx0.510$-selectable. Our positive results also apply to online matching on $1$-uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.

cs.DS↗

arXiv2024

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 (\`a 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

Engaging activities for teaching linear algebra

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum.

math.HO↗

arXiv2023

A Tight Competitive Ratio for Online Submodular Welfare Maximization

In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given $n$ bidders each equipped with a general (not necessarily monotone) submodular utility and $m$ items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of $\nicefrac{1}{4}$. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of $3-2\sqrt{2}\approx 0.171573 $ to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of $\approx 0.27493$, improving the previously known $\nicefrac{1}{4}$ guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest.

cs.DS↗

arXiv2022

Towards an Optimal Contention Resolution Scheme for Matchings

In this paper, we study contention resolution schemes for matchings. Given a fractional matching $x$ and a random set $R(x)$ where each edge $e$ appears independently with probability $x_e$, we want to select a matching $M \subseteq R(x)$ such that $\Pr[e \in M \mid e \in R(x)] \geq c$, for $c$ as large as possible. We call such a selection method a $c$-balanced contention resolution scheme. Our main results are (i) an asymptotically (in the limit as $\|x\|_\infty$ goes to 0) optimal $\simeq 0.544$-balanced contention resolution scheme for general matchings, and (ii) a $0.509$-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.

cs.DS↗

arXiv2022

Secretary Problems: The Power of a Single Sample

In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers $X_i$ that come from distributions $\mathcal{D}_i$, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample $Y_i$ from each distribution $\mathcal{D}_i$. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem was first studied by Correa et al. [SODA 2020] who managed to construct an algorithm that achieves a probability of $0.4529$. In this paper, we improve this probability to $0.5009$, almost matching an upper bound of $\simeq 0.5024$ which we show follows from earlier work. We also show that there is an algorithm which achieves the probability of $\simeq 0.5024$ asymptotically if no particular distribution is especially likely to yield the largest number. For the adversarial order version of the problem, we show that we can select the maximum number with a probability of $1/4$, and that this is best possible. Our work demonstrates that unlike in the case of the expected value objective studied by Rubinstein et al. [ITCS 2020], knowledge of a single sample is not enough to recover the factor of success guaranteed by full knowledge of the distribution.

cs.DS↗

arXiv2022

The Secretary Problem with Distributions

We study a variant of the secretary problem where candidates come from independent, not necessarily identical distributions known to us, and show that we can do at least as well as in the IID setting. This resolves a conjecture of Esfandiari et al.

cs.DS↗