AI 中文总结
本文给出无容量设施选址多胞形强、弱松弛体积的精确公式及一致渐近，揭示体积比在m~ln n处由超线性转为线性增长的尖锐相变。
AI 中文摘要
Lee、Skipper和Speakman（2018）提出了关于具有$m$个设施和$n$个客户的无容量设施选址多胞形的强松弛与弱松弛$\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n}$和$\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n}$的体积的公式或渐近表达式的问题（他们的问题3，\"Texas Hot\"）。我们给出了对所有$m$和$n$的精确公式，并给出了当$m+n\to\infty$时一致成立的渐近结果，无论$m$和$n$的相对增长速度如何。体积比满足$Vol(\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n})/Vol(\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n})\sim \prod_{j=2}^{n}\mleft(1+\tfrac{1}{(m-1)j}\mright)^{m}$。由此，我们读出了$m$的每个增长区域中的行为，并识别出在$m\asymp\ln n$处的尖锐转变。在转变之下，在强松弛的均匀随机点中，每个设施变量以高概率接近1，且该比率在$n$上超线性增长。在转变之上，在任一松弛的均匀随机点中，每个设施变量在$[0,1]$上渐近均匀，且该比率在$n$上线性增长。
英文摘要
Lee, Skipper, and Speakman (2018) asked for formulae or asymptotic expressions for the volumes of the strong and weak relaxations $\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n}$ and $\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n}$ of the uncapacitated facility-location polytope with $m$ facilities and $n$ customers (their Problem~3, ``Texas Hot''). We give exact formulae for all $m$ and $n$, and we give asymptotics that hold uniformly as $m+n\to\infty$, whatever the relative growth of $m$ and $n$. The volume ratio satisfies $Vol(\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n})/Vol(\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n})\sim \prod_{j=2}^{n}\mleft(1+\tfrac{1}{(m-1)j}\mright)^{m}$. From this, we read off the behavior in every growth regime of $m$, and we identify a sharp transition at $m\asymp\ln n$. Below it, in a uniformly random point of the strong relaxation, each facility variable is close to $1$ with high probability, and the ratio grows superlinearly in $n$. Above it, in a uniformly random point of either relaxation, each facility variable is asymptotically uniform on $[0,1]$, and the ratio grows linearly in $n$.
