Pasin Manurangsi
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We study the fine-grained complexity of the famous $k$-center problem in the metric induced by a graph with $n$ vertices and $m$ edges. The problem is NP-hard to approximate within a factor strictly better than $2$, and several $2$-approximation algorithms are known.
Two of the most well-known approaches for the $2$-approximation are (1) finding a maximal distance $r$-independent set (where the minimum pairwise distance is greater than $r$) and (2) Gonzalez's algorithm that iteratively adds the center farthest from the currently chosen centers.
For the approach based on distance-$r$ independent sets, Thorup [SIAM J. Comput. '05] already gave a nearly linear time algorithm. While Thorup's algorithm is not complicated, it still requires tools such as an approximate oracle for neighborhood size by Cohen [J. Comput. Syst. Sci. '97]. Our main result is a nearly straightforward algorithm that improves the running time by an $O(\log n$) factor. It results in an $(2+\eps)$-approximation for $k$-center in $O((m + n \log n)\log n \log(n/\eps))$ time.
For Gonzalez's algorithm [Theor. Comput. Sci. 85], we show that the simple $\widetilde{O}(mk)$-time implementation is nearly optimal if we insist the {\em exact} implementation. On the other hand, we show that an $(1+\eps)$-approximate version of the algorithm is efficiently implementable, leading to an $(2+\eps)$-approximation algorithm in running time $O((m + n \log n)\log^2 n / \eps)$. We also show that, unlike in the distance $r$-independent set-based algorithm, the dependency of $1/\eps$ in the running time is essentially optimal for $(1 + \eps)$-approximate Gonzalez's.
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Improved Inapproximability of VC Dimension and Littlestone’s Dimension via (Unbalanced) Biclique
ITCS 2023 (to appear)
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We study the complexity of computing (and approximating) VC Dimension and Littlestone's Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC Dimension and Littlestone's Dimension. With this connection, we derive a range of hardness of approximation results and running time lower bounds. For example, under the (randomized) Gap-Exponential Time Hypothesis or the Strongish Planted Clique Hypothesis, we show a tight inapproximability result: both dimensions are hard to approximate to within a factor of o(log n) in polynomial-time. These improve upon constant-factor inapproximability results from [Manurangsi and Rubinstein, COLT 2017].
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Fairness and privacy are two important concerns in social decision-making processes such as resource allocation. We study privacy in the fair allocation of indivisible resources using the well-established framework of differential privacy. We present algorithms for approximate envy-freeness and proportionality when two instances are considered to be adjacent if they differ only on the utility of a single agent for a single item. On the other hand, we provide strong negative results for both fairness criteria when the adjacency notion allows the entire utility function of a single agent to change.
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Differential privacy is often applied with a privacy parameter that is larger than the theory suggests is ideal; various informal justifications for tolerating large privacy parameters have been proposed.
In this work, we consider partial differential privacy (DP), which allows quantifying the privacy guarantee on a per-attribute basis.
In this framework, we study several basic data analysis and learning tasks, and design algorithms whose per-attribute privacy parameter is smaller that the best possible privacy parameter for the entire record of a person (i.e., all the attributes).
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Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds
Jelani Osei Nelson
Justin Y. Chen
Shyam Narayanan
Yinzhan Xu
SODA 2023 (to appear)
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We study the problem of releasing the weights of all-pairs shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most 1 in the ℓ1-distance. We give an algorithm with additive error ̃O(n^2/3/ε) in the ε-DP case and an algorithm with additive error ̃O(√n/ε) in the (ε, δ)-DP case, where n denotes the number of vertices. This positively answers a question of Sealfon [Sea16, Sea20], who asked whether a o(n) error algorithm exists. We also show that an additive error of Ω(n1/6) is necessary for any sufficiently small ε, δ > 0.
Furthermore, we show that if the graph is promised to have reasonably bounded weights, one can improve the error further to roughly n^{(√17−3)/2+o(1)}/ε in the ε-DP case and roughly n^{√2−1+o(1)}/ε in the (ε, δ)-DP case. Previously, it was only known how to obtain ̃O(n2/3/ε1/3) additive error in the ε-DP case and ̃O(√n/ε) additive error in the (ε, δ)-DP case for bounded-weight graphs [Sea16].
Finally, we consider a relaxation where a multiplicative approximation is allowed. We show that, with a multiplicative approximation factor k, the additive error can be reduced to ̃O(n^{1/2+O(1/k)}/ε) in the ε-DP case and ̃O(n^{1/3+O(1/k)}/ε) in the (ε, δ)-DP case.
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In this work, we study the task of estimating the numbers of distinct and k-occurring items in a time window under the constraint of differential privacy (DP). We consider several variants depending on whether the queries are on general time windows (between times t1 and t2), or are restricted to being cumulative (between times 1 and t2), and depending on whether the DP neighboring relation is event-level or the more stringent item-level. We obtain nearly tight upper and lower bounds on the errors of DP algorithms for these problems. En route, we obtain an event-level DP algorithm for estimating, at each time step, the number of distinct items seen over the last W updates with error polylogarithmic in W; this answers an open question of Bolot et al. (ICDT 2013).
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Leveraging Bias-Variance Trade-offs for Regression with Label Differential Privacy
Ashwinkumar Badanidiyuru Varadaraja
Avinash Varadarajan
Chiyuan Zhang
Ethan Leeman
Pritish Kamath
NeurIPS 2023 (2023)
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We propose a new family of label randomization mechanisms for the task of training regression models under the constraint of label differential privacy (DP). In particular, we leverage the trade-offs between bias and variance to construct better noising mechanisms depending on a privately estimated prior distribution over the labels. We demonstrate that these mechanisms achieve state-of-the-art privacy-accuracy trade-offs on several datasets, highlighting the importance of bias-reducing constraints when training neural networks with label DP. We also provide theoretical results shedding light on the structural properties of the optimal bias-reduced mechanisms.
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The Price of Justified Representation
Ayumi Igarashi
Edith Elkind
Piotr Faliszewski
Ulrike Schmidt-Kraepelin
Warut Suksompong
AAAI 2022
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In multiwinner approval voting, the goal is to select k-member committees based on voters' approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as coverage (maximizing the number of voters who approve at least one committee member) or social welfare (maximizing the number of approvals obtained by committee members). In this work, we investigate the impact of the JR axiom (as well as the more demanding EJR axiom) on social welfare and coverage. Our approach is threefold: we derive worst-case bounds on the loss of welfare/coverage that is caused by imposing JR, study the algorithmic complexity of finding 'good' committees that provide JR (obtaining a hardness result, an approximation algorithm, and a positive result for the one-dimensional setting), and study this problem empirically on several synthetic datasets.
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In this note, we consider the problem of differentially privately (DP) computing an anonymoized histogram, which is defined as the multiset of counts of the input dataset (without bucket labels). In the low-privacy regime ε ≥ 1, we give an ε-DP algorithm with an l1-error bound of O(√n/e^ε). In the high-privacy regime ε < 1, we give an Ω(sqrt(n log(1/ε)/ε)) lower bound on the l1 error. In both cases, our bounds asymptotically match the previously known lower/upper bounds due to [Suresh, NeurIPS 2019].
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We give the first polynomial time and sample (epsilon, delta)-differentially private (DP) algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers. Our algorithm succeeds for families of distributions that satisfy two well-studied properties in prior works on robust estimation: certifiable subgaussianity of directional moments and certifiable hypercontractivity of degree 2 polynomials. Our recovery guarantees hold in the “right affine-invariant norms”: Mahalanobis distance for mean, multiplicative spectral and relative Frobenius distance guarantees for covariance and injective norms for higher moments. Prior works obtained private robust algorithms for mean estimation of subgaussian distributions with bounded covariance. For covariance estimation, ours is the first efficient algorithm (even in the absence of outliers) that succeeds without any condition-number assumptions. Our algorithms arise from a new framework that provides a general blueprint for modifying convex relaxations for robust estimation to satisfy strong worst-case stability guarantees in the appropriate parameter norms whenever the algorithms produce witnesses of correctness in their run. We verify such guarantees for a modification of standard sum-of-squares (SoS) semidefinite programming relaxations for robust estimation. Our privacy guarantees are obtained by combining stability guarantees with a new “estimate dependent” noise injection mechanism in which noise scales with the eigenvalues of the estimated covariance. We believe this framework will be useful more generally in obtaining DP counterparts of robust estimators. Independently of our work, Ashtiani and Liaw [AL21] also obtained a polynomial time and sample private robust estimation algorithm for Gaussian distributions.
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