随机数划分问题中的数学障碍
Algorithmic Obstructions in the Random Number Partitioning Problem
摘要：
我们考虑了寻找数字划分问题（NPP）的近最优解的算法问题. NPP出现在许多应用中，包括随机对照试验的设计，多处理器调度和密码学;并且也具有理论意义。它具有一个所谓的计算间隙：当它的输入$X$具有分布$\mathcal{N}(0,I_n)$时，它的最优值是$Θ(\sqrt{n}2^{-n})$ w. h. p.;而最好的多项式时间算法实现的目标值只有$2^{-Θ(\log^2 n)}$，在本文中，我们开始研究这个差距的性质。受统计物理学见解的启发，我们研究了NPP的景观，并建立了重叠间隙属性（OGP）的存在，这是一种复杂的几何属性，被认为是大型算法的算法硬度的严格证据。通过利用OGP，我们建立了（a）任何足够稳定的算法，适当定义，都无法找到能量低于$2^{-ω(n \log^{-1/5} n)}$的近最优解;以及（b）非常自然的MCMC动态无法找到近最优解。我们的模拟表明，最先进的算法实现$2^{-Θ(\log^2 n)}$确实是稳定的，但正式验证这是一个开放的问题。OGP认为解的$m-$元组的重叠结构达到了一定的客观值。当$m$恒定时，我们证明了OGP在$2^{-Θ(n)}$体系中的存在，以及在$2^{-o(n)}$体系中的不存在。有趣的是，虽然，通过考虑与$m$不断增长的价值重叠，我们证明了存在的OGP的水平$2^{-ω(\sqrt{n\log n})}$。我们的失败的稳定算法的值$2^{-ω(n \log^{-1/5} n)}$的证明采用拉姆齐理论的极值组合学的方法，是独立的兴趣。
英文摘要：
We consider the algorithmic problem of finding a near-optimal solution for the number partitioning problem (NPP). The NPP appears in many applications, including the design of randomized controlled trials, multiprocessor scheduling, and cryptography; and is also of theoretical significance. It possesses a so-called statistical-to-computational gap: when its input $X$ has distribution $\mathcal{N}(0,I_n)$, its optimal value is $Θ(\sqrt{n}2^{-n})$ w.h.p.; whereas the best polynomial-time algorithm achieves an objective value of only $2^{-Θ(\log^2 n)}$, w.h.p. In this paper, we initiate the study of the nature of this gap. Inspired by insights from statistical physics, we study the landscape of NPP and establish the presence of the Overlap Gap Property (OGP), an intricate geometric property which is known to be a rigorous evidence of an algorithmic hardness for large classes of algorithms. By leveraging the OGP, we establish that (a) any sufficiently stable algorithm, appropriately defined, fails to find a near-optimal solution with energy below $2^{-ω(n \log^{-1/5} n)}$; and (b) a very natural MCMC dynamics fails to find near-optimal solutions. Our simulations suggest that the state of the art algorithm achieving $2^{-Θ(\log^2 n)}$ is indeed stable, but formally verifying this is left as an open problem. OGP regards the overlap structure of $m-$tuples of solutions achieving a certain objective value. When $m$ is constant we prove the presence of OGP in the regime $2^{-Θ(n)}$, and the absence of it in the regime $2^{-o(n)}$. Interestingly, though, by considering overlaps with growing values of $m$ we prove the presence of the OGP up to the level $2^{-ω(\sqrt{n\log n})}$. Our proof of the failure of stable algorithms at values $2^{-ω(n \log^{-1/5} n)}$ employs methods from Ramsey Theory from the extremal combinatorics, and is of independent interest.
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