不完全偏好表的随机流行匹配
Random Popular Matchings with Incomplete Preference Lists
摘要：
给定一组$n$人的$A$和一组$m \geq n$项目的$B$，每个人都有一个按照偏好顺序排列其偏好项目的列表，我们希望将每个人与唯一的项目进行匹配。一个匹配的$M$被称为流行的，如果对于任何其他匹配的$M'$，更喜欢$M$的人数不少于更喜欢$M'$的人数。对于给定的$n$和$m$，考虑当每个人的偏好列表独立且均匀地随机生成时，流行匹配的存在概率。在此之前，Mahdian证明了当人们的偏好列表是严格的（不包含任何关系）和完整的（包含$B$中的所有项目）时，如果$α= m/n > α_*$，其中$α_* \approx 1.42$是方程$x^2 = e^{1/x}$的根，那么流行匹配以概率$1-o(1)$存在;如果$α< α_*$，那么流行匹配以概率$o(1)$存在，即在$α_*$发生相变。在本文中，我们研究的情况下，人们的偏好列表是严格的，但不完整的相变。我们表明，在每个人都有一个恒定的$k \geq 4$长度的偏好列表的情况下，类似的相变发生在$α_k$，其中$α_k \geq 1$是方程$x e^{-1/2x} = 1-(1-e^{-1/x})^{k-1}$的根。
英文摘要：
Given a set $A$ of $n$ people and a set $B$ of $m \geq n$ items, with each person having a list that ranks his/her preferred items in order of preference, we want to match every person with a unique item. A matching $M$ is called popular if for any other matching $M'$, the number of people who prefer $M$ to $M'$ is not less than the number of those who prefer $M'$ to $M$. For given $n$ and $m$, consider the probability of existence of a popular matching when each person's preference list is independently and uniformly generated at random. Previously, Mahdian showed that when people's preference lists are strict (containing no ties) and complete (containing all items in $B$), if $α= m/n > α_*$, where $α_* \approx 1.42$ is the root of equation $x^2 = e^{1/x}$, then a popular matching exists with probability $1-o(1)$; and if $α< α_*$, then a popular matching exists with probability $o(1)$, i.e. a phase transition occurs at $α_*$. In this paper, we investigate phase transitions in the case that people's preference lists are strict but not complete. We show that in the case where every person has a preference list with length of a constant $k \geq 4$, a similar phase transition occurs at $α_k$, where $α_k \geq 1$ is the root of equation $x e^{-1/2x} = 1-(1-e^{-1/x})^{k-1}$.
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。
