Special Issue on Selected Papers from the 12th International Conference and Workshops on Algorithms and Computation, WALCOM 2018 Random Popular Matchings with Incomplete Preference Lists Suthee Ruangwises and Toshiya Itoh Vol. 23, no. 5, pp. 815-835, 2019. Regular paper. Abstract 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[Mahdian, Conf. El. Comm., 2006] showed that when people's preference lists are strict (containing no ties) and complete (containing all items in $B$), if $\alpha = m/n > \alpha_*$, where $\alpha_* \approx 1.42$ is the root of equation $x^2 = e^{1/x}$, then a popular matching exists with probability $1-o(1)$; and if $\alpha < \alpha_*$, then a popular matching exists with probability $o(1)$, i.e. a phase transition occurs at $\alpha_*$. 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 $\alpha_k$, where $\alpha_k \geq 1$ is the root of equation $x e^{-1/2x} = 1-(1-e^{-1/x})^{k-1}$. Submitted: March 2018. Reviewed: April 2019. Revised: April 2019. Accepted: June 2019. Final: September 2019. Published: October 2019. Communicated by M. Sohel Rahman, Wing-Kin Sung, and Ryuhei Uehara article (PDF) BibTeX