光滑分析的Pareto最优解
Pareto Optimal Solutions for Smoothed Analysts
摘要：
考虑具有$n$二元变量和$d+1$线性目标函数的优化问题。每一个有效解$x \in \{0,1\}^n$都会在$\R^{d+1}$中产生一个目标向量，并且人们经常想要枚举其中的帕累托最优解。在最坏的情况下，可能有指数多帕累托最优;然而，最近表明，在（推广）的平滑分析框架，预期的数量是多项式在$n$。不幸的是，所得到的界限对$d$有相当差的依赖性;大约是$n^{d^d}$。在本文中，我们显示了一个显着改善的界的$n^{2d}$。我们的证明是基于分析两个算法。第一种算法在输入帕累托最优$x$时输出一个“证据”，其中包含关于$x$的目标向量、$x$的坐标以及$x$的目标向量所在的空间区域$B$的线索。第二个算法可以被看作是第一个算法的{ \em推测}执行--它可以从证词的线索和概率空间的结果中的\emph{some}唯一地重建$x$。概率空间的结果的其余部分刚好足以限制$x$的目标向量落入区域$B$的概率。
英文摘要：
Consider an optimization problem with $n$ binary variables and $d+1$ linear objective functions. Each valid solution $x \in \{0,1\}^n$ gives rise to an objective vector in $\R^{d+1}$, and one often wants to enumerate the Pareto optima among them. In the worst case there may be exponentially many Pareto optima; however, it was recently shown that in (a generalization of) the smoothed analysis framework, the expected number is polynomial in $n$. Unfortunately, the bound obtained had a rather bad dependence on $d$; roughly $n^{d^d}$. In this paper we show a significantly improved bound of $n^{2d}$. Our proof is based on analyzing two algorithms. The first algorithm, on input a Pareto optimal $x$, outputs a "testimony" containing clues about $x$'s objective vector, $x$'s coordinates, and the region of space $B$ in which $x$'s objective vector lies. The second algorithm can be regarded as a {\em speculative} execution of the first -- it can uniquely reconstruct $x$ from the testimony's clues and just \emph{some} of the probability space's outcomes. The remainder of the probability space's outcomes are just enough to bound the probability that $x$'s objective vector falls into the region $B$.
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