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Novel algorithms for maximum DS decomposition
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2020-12-31 , DOI: 10.1016/j.tcs.2020.12.041
Shengminjie Chen , Wenguo Yang , Suixiang Gao , Rong Jin

DS decomposition plays an important role in set function optimization problem, because there is DS decomposition for any set function. How to design an efficient and effective algorithm to solve maximizing DS decomposition is a heated problem. In this work, we propose a framework called Parameter Conditioned Greedy Algorithm which has a deterministic version and two random versions. In more detail, this framework uses the difference with parameter decomposition function and combines non-negative condition. Besides, if we set the different parameters, the framework can return solution with different approximation ratio. Also, we choose two special case to show our deterministic algorithm gets f(Sk)(e1cg)g(Sk)(1e1)[f(OPT)g(OPT)] and f(Sk)(1cg)g(Sk)(1e1)f(OPT)g(OPT) respectively for cardinality constrained problem, where cg is the curvature of monotone submodular set function. To speed the deterministic algorithm, we introduce a random sample set whose intersection with the optimal solution is as nonempty as possible. Importantly, it also can get the same approximation ratio as deterministic algorithm under expectation. Further, for maximization DS decomposition without constraint, our another random algorithm gets E[f(Sk)(e1cg)g(Sk)](1e1)[f(OPT)g(OPT)] and E[f(Sk)(1cg)g(Sk)](1e1)f(OPT)g(OPT) respectively. Because the Parameter Conditioned Algorithm is the general framework, different users can choose the parameters that fit their problem to get a better approximation.



中文翻译:

用于最大DS分解的新颖算法

DS分解在集合函数优化问题中起重要作用,因为任何集合函数都有DS分解。如何设计一种有效且有效的算法来解决DS分解最大化问题是一个亟待解决的问题。在这项工作中,我们提出了一个称为参数条件贪婪算法的框架,该框架具有确定性版本和两个随机版本。更详细地,该框架将差异与参数分解功能结合使用,并结合了非负条件。此外,如果我们设置不同的参数,则框架可以返回具有不同近似比率的解。另外,我们选择两种特殊情况来展示确定性算法F小号ķ-Ë-1个-CGG小号ķ1个-Ë-1个[FØPŤ-GØPŤ]F小号ķ-1个-CGG小号ķ1个-Ë-1个FØPŤ-GØPŤ 分别针对基数约束问题,其中 CG是单调子模集函数的曲率。为了加快确定性算法的速度,我们引入了一个随机样本集,该样本集与最优解的交点应尽可能为非空。重要的是,在期望的情况下,它也可以获得与确定性算法相同的近似率。此外,为了不受限制地最大化DS分解,我们的另一种随机算法Ë[F小号ķ-Ë-1个-CGG小号ķ]1个-Ë-1个[FØPŤ-GØPŤ]Ë[F小号ķ-1个-CGG小号ķ]1个-Ë-1个FØPŤ-GØPŤ分别。由于参数条件算法是通用框架,因此不同的用户可以选择适合其问题的参数以获得更好的近似值。

更新日期:2021-01-22
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