arXiv subjects

Aravind Srinivasan

Publications and source records attributed to Aravind Srinivasan.

At least 19 recordsLinked to original sources

arXiv2026

Concentration from Product Moments via an Additional Element of Randomness

The standard method of exponential moments for proving concentration bounds can often be replaced by an argument based on elementary symmetric polynomials. We introduce an additional element of randomness into this framework, which reduces the problem to bounding product moments over a uniformly sampled set of indices. We show that this approach gives useful bounds in three settings. For read-$Δ$ families under limited independence, we obtain bounds governed by the degrees of randomly induced dependency subgraphs, improving the dependence on worst-case degrees. For random binary linear hashing with (semi-)random inputs, we derive fixed-bin and maximum-load bounds by controlling the rank defect of random tuples of input keys. Finally, for stochastic processes, we show how decay of product moments yields concentration bounds, recovering the spectral and mixing-time scales for finite-state Markov chains.

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arXiv2026

Differentially Private Graph Coloring

Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of computing an edge-differentially private vertex coloring. In this paper, we present two novel algorithms for this problem. Both algorithms begin by coloring each vertex uniformly at random from a fixed-size palette, and then apply the exponential mechanism to locally resample colors for either all vertices or a selected subset of vertices. Any non-trivial edge differentially private coloring of a graph needs to be defective, as a proper coloring exposes the non-existence of an edge between two vertices of the same color. A coloring is $k$-defective if each vertex shares its color with at most $k$ of its neighbors. Our goal is to design coloring algorithms that use the minimum number of colors, while achieving the smallest possible defect under the edge-differential privacy. Our first algorithm applies to $d$-inductive graphs with maximum degree $Δ$. We show that it yields a $3ε$-differentially private coloring with $O(\frac{\log n}ε+d)$ maximum defect, using a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$. Our second algorithm utilizes noisy thresholding to guarantee $O(\frac{\log n}ε)$ maximum defect, using a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$, generalizing the results to all graphs rather than just $d$-inductive ones.

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arXiv2026

Improved Guarantees for Offline Stochastic Matching via New Ordered Contention Resolution Schemes

Matching is one of the most fundamental and broadly applicable problems across many domains. In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models. Here, each edge in the graph has a known, independent probability of existing derived from some prediction. Algorithms must probe edges to determine existence and match them irrevocably if they exist. Further, each vertex may have a patience constraint denoting how many of its neighboring edges can be probed. We present new ordered contention resolution schemes yielding improved approximation guarantees for some of the foundational problems studied in this area. For stochastic matching with patience constraints in general graphs, we provide a 0.382-approximate algorithm assuming each vertex has patience at least $2$. Under this assumption, we improve upon the previous best 0.31-approximation of Baveja et al. (2018). When the vertices do not have patience constraints, we describe a 0.432-approximate random order probing algorithm with several corollaries such as an improved guarantee for the Prophet Secretary problem under Edge Arrivals. Finally, for the special case of bipartite graphs with unit patience constraints on one of the partitions, we show a 0.632-approximate algorithm that improves on the recent $1/3$-guarantee of Hikima et al. (2021).

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arXiv2025

Dimension-Free Correlated Sampling for the Hypersimplex

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $α$ times their input vectors' $\ell_1$ distance. A value of $α=O(\log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(\log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $\log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.

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arXiv2025

Controlling the Spread of Epidemics on Networks with Differential Privacy

Designing effective strategies for controlling epidemic spread by vaccination is an important question in epidemiology, especially in the early stages when vaccines are limited. This is a challenging question when the contact network is very heterogeneous, and strategies based on controlling network properties, such as the degree and spectral radius, have been shown to be effective. Implementation of such strategies requires detailed information on the contact structure, which might be sensitive in many applications. Our focus here is on choosing effective vaccination strategies when the edges are sensitive and differential privacy guarantees are needed. Our main contributions are $(\varepsilon,δ)$-differentially private algorithms for designing vaccination strategies by reducing the maximum degree and spectral radius. Our key technique is a private algorithm for the multi-set multi-cover problem, which we use for controlling network properties. We evaluate privacy-utility tradeoffs of our algorithms on multiple synthetic and real-world networks, and show their effectiveness.

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arXiv2024

Proportionally Fair Matching via Randomized Rounding

Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond to demographic groups or other protected traits where we seek to ensure roughly equal representation from each group. It is known that, assuming ETH, it is impossible to approximate the problem with $\ell$ colors in time $2^{o(\ell)} n^{\mathcal{O}(1)}$ (i.e., subexponential in $\ell$) even on \emph{unweighted path graphs}. Further, even determining the existence of a non-empty matching satisfying proportionality is NP-Hard. To overcome this hardness, we relax the stringent proportional fairness constraints to a probabilistic notion. We introduce a notion we call $δ$-\textsc{ProbablyAlmostFair}, where we ensure proportionality up to a factor of at most $(1 \pm δ)$ for some small $δ>0$ with high probability. The violation $δ$ can be brought arbitrarily close to $0$ for some \emph{good} instances with large values of matching size. We propose and analyze simple and fast algorithms for bipartite graphs that achieve constant-factor approximation guarantees, and return a $δ$-\textsc{ProbablyAlmostFair} matching.

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arXiv2024

Online Dependent Rounding Schemes for Bipartite Matchings, with Applications

We introduce the abstract problem of rounding an unknown fractional bipartite $b$-matching $\bf{x}$ revealed online (e.g., output by an online fractional algorithm), exposed node-by-node on~one~side. The objective is to maximize the \emph{rounding ratio} of the output matching $M$, which is the minimum over all fractional $b$-matchings $\bf{x}$, and edges $e$, of the ratio $\Pr[e\in M]/x_e$. In analogy with the highly influential offline dependent rounding schemes of Gandhi et al.~(FOCS'02, JACM'06), we refer to such algorithms as \emph{online dependent rounding schemes} (ODRSes). This problem, with additional restrictions on the possible inputs $\bf{x}$, has played a key role in recent developments in online computing. We provide the first generic $b$-matching ODRSes that impose no restrictions on $\bf{x}$. Specifically, we provide ODRSes with rounding ratios of $0.646$ and $0.652$ for $b$-matchings and simple matchings, respectively. This breaks the natural barrier of $1-1/e$, prevalent for online matching problems, and numerous online problems more broadly. Using our ODRSes, we provide a number of algorithms with similar better-than-$(1-1/e)$ ratios for several problems in online edge coloring, stochastic optimization, and more. Our techniques, which have already found applications in several follow-up works (Patel and Wajc SODA'24, Blikstad et al.~SODA'25, Braverman et al.~SODA'25, and Aouad et al.~2024), include periodic use of \emph{offline} contention resolution schemes (in online algorithm design), grouping nodes, and a new scaling method which we call \emph{group discount and individual markup}.

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arXiv2024

Concentration of Submodular Functions and Read-k Families Under Negative Dependence

We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in (Qiu and Singla [QS22]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (Chekuri, Vondrak, and Zenklusen [CVZ10] and Harvey and Olver [HO14]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Gav+15] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Gav+15].

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arXiv2024

Barter Exchange with Shared Item Valuations

In barter exchanges agents enter seeking to swap their items for other items on their wishlist. We consider a centralized barter exchange with a set of agents and items where each item has a positive value. The goal is to compute a (re)allocation of items maximizing the agents' collective utility subject to each agent's total received value being comparable to their total given value. Many such centralized barter exchanges exist and serve crucial roles; e.g., kidney exchange programs, which are often formulated as variants of directed cycle packing. We show finding a reallocation where each agent's total given and total received values are equal is NP-hard. On the other hand, we develop a randomized algorithm that achieves optimal utility in expectation and where, i) for any agent, with probability 1 their received value is at least their given value minus $v^*$ where $v^*$ is said agent's most valuable owned and wished-for item, and ii) each agent's given and received values are equal in expectation.

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arXiv2024

Dependent randomized rounding for clustering and partition systems with knapsack constraints

Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among the cluster-centers. This, and much more general clustering problems, can be formulated with "knapsack" and "partition" constraints. We develop new randomized algorithms targeting such problems, and study two in particular: multi-knapsack median and multi-knapsack center. Our rounding algorithms give new approximation and pseudo-approximation algorithms for these problems. One key technical tool, which may be of independent interest, is a new tail bound analogous to Feige (2006) for sums of random variables with unbounded variances. Such bounds can be useful in inferring properties of large networks using few samples.

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arXiv2024

Stochastic Optimization and Learning for Two-Stage Supplier Problems

The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. In addition to the standard (homogeneous) setting where all clients must be within a distance $R$ of the nearest facility, we provide results for the more general problem where the radius demands may be inhomogeneous (i.e., different for each client). We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints, and provide results for these settings. We derive results for the most general distributional setting, where there is only black-box access to the underlying distribution. To accomplish this, we first develop algorithms for the polynomial scenarios setting; we then employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.

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arXiv2023

Planning to Fairly Allocate: Probabilistic Fairness in the Restless Bandit Setting

Restless and collapsing bandits are often used to model budget-constrained resource allocation in settings where arms have action-dependent transition probabilities, such as the allocation of health interventions among patients. However, state-of-the-art Whittle-index-based approaches to this planning problem either do not consider fairness among arms, or incentivize fairness without guaranteeing it. We thus introduce ProbFair, a probabilistically fair policy that maximizes total expected reward and satisfies the budget constraint while ensuring a strictly positive lower bound on the probability of being pulled at each timestep. We evaluate our algorithm on a real-world application, where interventions support continuous positive airway pressure (CPAP) therapy adherence among patients, as well as on a broader class of synthetic transition matrices. We find that ProbFair preserves utility while providing fairness guarantees.

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arXiv2023

Online Matching Frameworks under Stochastic Rewards, Product Ranking, and Unknown Patience

We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm when each item has its own hazard rate for making the customer depart, and a 1/2-approximate algorithm when the customer has a general item-independent patience distribution. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. We complement these positive results by presenting three additional negative results relating to these problems.

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arXiv2023

Rawlsian Fairness in Online Bipartite Matching: Two-sided, Group, and Individual

Online bipartite-matching platforms are ubiquitous and find applications in important areas such as crowdsourcing and ridesharing. In the most general form, the platform consists of three entities: two sides to be matched and a platform operator that decides the matching. The design of algorithms for such platforms has traditionally focused on the operator's (expected) profit. Since fairness has become an important consideration that was ignored in the existing algorithms a collection of online matching algorithms have been developed that give a fair treatment guarantee for one side of the market at the expense of a drop in the operator's profit. In this paper, we generalize the existing work to offer fair treatment guarantees to both sides of the market simultaneously, at a calculated worst case drop to operator profit. We consider group and individual Rawlsian fairness criteria. Moreover, our algorithms have theoretical guarantees and have adjustable parameters that can be tuned as desired to balance the trade-off between the utilities of the three sides. We also derive hardness results that give clear upper bounds over the performance of any algorithm.

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arXiv2023

Fair Clustering Under a Bounded Cost

Clustering is a fundamental unsupervised learning problem where a dataset is partitioned into clusters that consist of nearby points in a metric space. A recent variant, fair clustering, associates a color with each point representing its group membership and requires that each color has (approximately) equal representation in each cluster to satisfy group fairness. In this model, the cost of the clustering objective increases due to enforcing fairness in the algorithm. The relative increase in the cost, the ''price of fairness,'' can indeed be unbounded. Therefore, in this paper we propose to treat an upper bound on the clustering objective as a constraint on the clustering problem, and to maximize equality of representation subject to it. We consider two fairness objectives: the group utilitarian objective and the group egalitarian objective, as well as the group leximin objective which generalizes the group egalitarian objective. We derive fundamental lower bounds on the approximation of the utilitarian and egalitarian objectives and introduce algorithms with provable guarantees for them. For the leximin objective we introduce an effective heuristic algorithm. We further derive impossibility results for other natural fairness objectives. We conclude with experimental results on real-world datasets that demonstrate the validity of our algorithms.

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arXiv2022

Improved Bi-point Rounding Algorithms and a Golden Barrier for $k$-Median

The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better than $2.588$ under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open $k+o(k)$ facilities. This gives a barrier to current approaches for obtaining an approximation better than $2 \sqrtϕ \approx 2.544$. Altogether we reduce the approximation gap of bi-point solutions by two thirds.

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arXiv2022

Fair Disaster Containment via Graph-Cut Problems

Graph cut problems are fundamental in Combinatorial Optimization, and are a central object of study in both theory and practice. Furthermore, the study of \emph{fairness} in Algorithmic Design and Machine Learning has recently received significant attention, with many different notions proposed and analyzed for a variety of contexts. In this paper we initiate the study of fairness for graph cut problems by giving the first fair definitions for them, and subsequently we demonstrate appropriate algorithmic techniques that yield a rigorous theoretical analysis. Specifically, we incorporate two different notions of fairness, namely \emph{demographic} and \emph{probabilistic individual} fairness, in a particular cut problem that models disaster containment scenarios. Our results include a variety of approximation algorithms with provable theoretical guarantees.

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arXiv2022

Controlling Epidemic Spread using Probabilistic Diffusion Models on Networks

The spread of an epidemic is often modeled by an SIR random process on a social network graph. The MinINF problem for optimal social distancing involves minimizing the expected number of infections, when we are allowed to break at most $B$ edges; similarly the MinINFNode problem involves removing at most $B$ vertices. These are fundamental problems in epidemiology and network science. While a number of heuristics have been considered, the complexity of these problems remains generally open. In this paper, we present two bicriteria approximation algorithms for MinINF, which give the first non-trivial approximations for this problem. The first is based on the cut sparsification result of Karger \cite{karger:mathor99}, and works when the transmission probabilities are not too small. The second is a Sample Average Approximation (SAA) based algorithm, which we analyze for the Chung-Lu random graph model. We also extend some of our results to tackle the MinINFNode problem.

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