Pasin Manurangsi
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We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that the EF1 existence can be guaranteed even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are n agents and the number of goods m is divisible by the number of groups k, then an envy-free outcome exists with high probability if m = ω(log n), and this bound is tight. On the other hand, if m is not divisible by k, then an envy-free outcome is unlikely to exist as long as m = o(√n).
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We prove the following asymptotically tight lower bound for k-color discrepancy: For any k ≥ 2, there exists a hypergraph with n vertices such that its k-color discrepancy is at least Ω(√n). This improves on the previously known lower bound of Ω(√n/ log k) due to Caragiannis et al. [CLS25]. As an application, we show that our result implies improved lower bounds for group fair division.
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Rank aggregation is a task of combining the rankings of items from multiple users into a single ranking that best represents the users' rankings. Alabi et al. (AAAI'22) presents differentially-private (DP) polynomial-time approximation schemes (PTASes) and 5-approximation algorithms with certain additive errors for the Kemeny rank aggregation problem in both central and local models. In this paper, we present improved DP PTASes with smaller additive error in the central model. Furthermore, we are first to study the footrule rank aggregation problem under DP. We give a near-optimal algorithm for this problem; as a corollary, this leads to 2-approximation algorithms with the same additive error as the 5-approximation algorithms of Alabi et al. for the Kemeny rank aggregation problem in both central and local models.
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We consider a setting where we have a ground set ℳ together with real-valued set functions f₁, … , f_n, and the goal is to partition ℳ into two sets S₁,S₂ such that |f_i(S₁) - f_i(S₂)| is small for every i. Many results in discrepancy theory can be stated in this form with the functions f_i being additive. In this work, we initiate the study of the unstructured case where f_i is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most O(√{n log n}).
Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to O(√{n log n}) goods always exists for n agents with monotone utilities. Previously, only an O(n) bound was known for this setting.
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We study the d-dimensional knapsack problem. We are given a set of items, each with a d-dimensional cost vector and a profit, along with a d-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A polynomial-time approximation scheme (PTAS) with running time n^{Θ(d/{ε})} has long been known for this problem, where {ε} is the error parameter and n is the encoding size. Despite decades of active research, the best running time of a PTAS has remained O(n^{⌈ d/{ε} ⌉ - d}). Unfortunately, existing lower bounds only cover the special case with two dimensions d = 2, and do not answer whether there is a n^{o(d/({ε)})}-time PTAS for larger values of d.
In this work, we show that the running times of the best-known PTAS cannot be improved up to a polylogarithmic factor assuming the Exponential Time Hypothesis (ETH). Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions. Then, using a recent result of [Bafna Karthik and Minzer, STOC'25], we succeed in exhibiting tight trade-off between d and {ε} for all regimes of the parameters assuming d is sufficiently large. Informally, our result also shows that under ETH, for any function f there is no f(d/({ε)}) ⋅ n^{õ(d/({ε)})}-time (1-{ε})-approximation for d-dimensional knapsack, where n is the number of items and õ hides polylogarithmic factors in d/({ε)}.
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Differential privacy can be achieved in a distributed manner, where multiple parties add independent noise such that their sum protects the overall dataset with differential privacy. A common technique here is for each party to sample their noise from the decomposition of an infinitely divisible distribution. We introduce two novel mechanisms in this setting: 1) the generalized discrete Laplace (GDL) mechanism, whose distribution (which is closed under summation) follows from differences of i.i.d. negative binomial shares, and 2) The multi-scale discrete Laplace (MSDLap) mechanism, which follows the sum of multiple i.i.d. discrete Laplace shares at different scales. The mechanisms can be parameterized to have 𝑂(Δ^3𝑒^{−𝜀}) and 𝑂 (min(Δ^3𝑒^{−𝜀}, Δ^2𝑒^{−2𝜀/3})) MSE, respectively, where the latter bound matches known optimality results. Furthermore, the MSDLap mechanism has the optimal MSE including constants as 𝜀 → ∞. We also show a transformation from the discrete setting to the continuous setting, which allows us to transform both mechanisms to the continuous setting and thereby achieve the optimal 𝑂 (Δ^2𝑒^{−2𝜀/3}) MSE. To our knowledge, these are the first infinitely divisible additive noise mechanisms that achieve order-optimal MSE under pure differential privacy for either the discrete or continuous setting, so our work shows formally there is no separation in utility when query-independent noise adding mechanisms are restricted to infinitely divisible noise. For the continuous setting, our result improves upon Pagh and Stausholm’s Arete distribution which gives an MSE of 𝑂(Δ^2𝑒^{−𝜀/4}) [35]. We apply our results to improve a state of the art multi-message shuffle DP protocol from [3] in the high 𝜀 regime.
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Several resource allocation settings involve agents with unequal entitlements represented by weights. We analyze weighted fair division from an asymptotic perspective: if m items are divided among n agents whose utilities are independently sampled from a probability distribution, when is it likely that a fair allocation exist? We show that if the ratio between the weights is bounded, a weighted envy-free allocation exists with high probability provided that m = Ω(n log n/ log log n), generalizing a prior unweighted result. For weighted proportionality, we establish a sharp threshold of m = n/(1 − μ) for the transition from non-existence to existence, where μ ∈ (0, 1) denotes the mean of the distribution. In addition, we prove that for two agents, a weighted envy-free (and weighted proportional) allocation is likely to exist if m = ω(√r), where r denotes the ratio between the two weights.
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User-level differentially private stochastic convex optimization (DP-SCO) has garnered significant attention due to the paramount importance in safeguarding user privacy in large-scale machine learning applications. Current methods, such as those based on Differentially Private Stochastic Gradient Descent (DP-SGD), often struggle with high noise accumulation and suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a novel linear-time algorithm that leverages robust statistics, specifically the geometric median and trimmed mean, to overcome these challenges. Our approach uniquely bounds the sensitivity of all intermediate iterates of SGD with gradient estimation based on robust statistics, thereby significantly reducing the gradient estimation noise and enhancing the privacy-utility trade-off. By sidestepping the repeated privatization required by previous methods, our algorithm not only achieves an improved theoretical privacy-utility balance but also maintains computational efficiency. This work sets the stage for more robust and efficient privacy-preserving techniques in machine learning, with implications for future research and application in the field.
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We study the allocation of indivisible goods under conflicting constraints, represented by a graph. In this framework, vertices correspond to goods and edges correspond to conflicts between a pair of goods. Each agent is allocated an independent set in the graph. In a recent work of [Kumar et al., 2024], it was shown that a maximal EF1 allocation exists for interval graphs and two agents with monotone valuations. We significantly extend this result by establishing that a maximal EF1 allocation exists for *any graph* when the two agents have monotone valuations. To compute such an allocation, we present a polynomial-time algorithm for additive valuations as well as a pseudo-polynomial time algorithm for monotone valuations. Moreover, we complement our findings by providing a counter example demonstrating a maximal EF1 allocation may not exist for three agents with monotone valuations. Additionally, we establish NP-hardness of determining the existence of such allocations for every fixed number n of agents.
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We study differential privacy (DP) in a multi-party setting where each party only trusts a (known) subset of the other parties with its data. Specifically, given a trust graph where vertices correspond to parties and neighbors are mutually trusting, we give a DP algorithm for aggregation with a much better privacy-utility trade-off than in the well-studied local model of DP (where each party trusts no other party). We further study a robust variant where each party trusts all but an unknown subset of at most t of its neighbors (where t is a given parameter), and give an algorithm for this setting. We complement our algorithms with lower bounds, and discuss implications of our work to other tasks in private learning and analytics.
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