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Private non-monotone submodular maximization
Journal of Combinatorial Optimization ( IF 1 ) Pub Date : 2022-06-23 , DOI: 10.1007/s10878-022-00875-w
Xin Sun, Gaidi Li, Yapu Zhang, Zhenning Zhang

We propose a private algorithm for the problem of maximizing a submodular but not necessary monotone set function over a down-closed family of sets. The constraint is very general since it includes some important and typical constraints such as knapsack and matroid constraints. Our algorithm Differentially Private Measure Continuous Greedy is proved to be \({\mathcal {O}}(\epsilon )\)-differential private. For the multilinear relaxation of the above problem, it yields \(\left( Te^{-T}-o(1)\right) \)-approximation guarantee with additive error \({\mathcal {O}}\left( \frac{2\varDelta }{\epsilon n^4}\right) \), where \(T\in [0,1]\) is the stopping time of the algorithm, \(\varDelta \) is the defined sensitivity of the objective function, which is associated to a sensitive dataset, and n is the size of the given ground set. For a specific matroid constraint, we could obtain a discrete solution with near 1/e-approximation guarantee and same additive error by lossless rounding technique. Besides, our algorithm can be also applied in monotone case. The approximation guarantee is \(\left( 1-e^{-T}-o(1)\right) \) when the submodular set function is monotone. Furthermore, we give a conclusion in terms of the density of the relaxation constraint, which is always at least as good as the tight bound \((1-1/e)\).



中文翻译:

私有非单调子模最大化

我们提出了一种私有算法,用于在向下封闭的集合族上最大化子模但不是必需的单调集合函数。该约束非常通用,因为它包括一些重要且典型的约束,例如背包和拟阵约束。我们的算法Differentially Private Measure Continuous Greedy被证明是\({\mathcal {O}}(\epsilon )\) -差分私有。对于上述问题的多线性松弛,它产生\(\left( Te^{-T}-o(1)\right) \) -具有加性误差的近似保证\({\mathcal {O}}\left( \frac{2\varDelta }{\epsilon n^4}\right) \),其中\(T\in [0,1]\)是算法的停止时间,\(\varDelta \)是与敏感数据集相关联的目标函数的定义灵敏度,n是给定地面集的大小。对于特定的拟阵约束,我们可以通过无损舍入技术获得具有接近 1/ e近似保证和相同附加误差的离散解。此外,我们的算法也可以应用于单调情况。当子模集函数是单调的时,近似保证是\(\left( 1-e^{-T}-o(1)\right) \) 。此外,我们根据松弛约束的密度给出了一个结论,它总是至少与紧界\((1-1/e)\)一样好。

更新日期:2022-06-23
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