arXiv 1012.11632010-12-06cs.DS

光滑Pareto最优解个数的下界

Lower Bounds for the Smoothed Number of Pareto optimal Solutions

Tobias Brunsch, Heiko Roeglin

摘要：

在2009年，Roeglin和Teng证明了线性多目标优化问题的Pareto最优解的平滑数在变量数$n$和半随机输入模型的最大密度$ϕ$中是多项式有界的。然而，它们的界不是很实用，因为指数在目标函数的数量$d+1$中呈指数增长。在最近的一项突破中，Moitra和O 'Donnell将这一结合显著改进为$O(n^{2d} ϕ^{d(d+1)/2})$。Moitra和O'Donnell在他们的论文中提出的一个“有趣的问题”是，这个界限可以进一步改进多少。前面的下界不排除多项式上界的可能性，其次数不依赖于$d$。在本文中，我们解决了这个问题，通过构造一类实例，$Ω((n ϕ)^{(d-\log{d}) \cdot (1-Θ{1/ϕ})})$ Pareto最优解的期望。对于双准则的情况下，我们提出了一个更高的下限的$Ω(n^2 ϕ^{1 - Θ{1/ϕ}})$，这几乎匹配已知的上限的$O(n^2 ϕ)$。

英文摘要：

In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $ϕ$ of the semi-random input model for any fixed number of objective functions. Their bound is, however, not very practical because the exponents grow exponentially in the number $d+1$ of objective functions. In a recent breakthrough, Moitra and O'Donnell improved this bound significantly to $O(n^{2d} ϕ^{d(d+1)/2})$. An "intriguing problem", which Moitra and O'Donnell formulate in their paper, is how much further this bound can be improved. The previous lower bounds do not exclude the possibility of a polynomial upper bound whose degree does not depend on $d$. In this paper we resolve this question by constructing a class of instances with $Ω((n ϕ)^{(d-\log{d}) \cdot (1-Θ{1/ϕ})})$ Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of $Ω(n^2 ϕ^{1 - Θ{1/ϕ}})$, which almost matches the known upper bound of $O(n^2 ϕ)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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