Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unpopularity Factor in the Marriage and Roommates Problems   Given a set $A$ of $n$ people, with each person having a preference list that
ranks a subset of $A$ as his/her acceptable partners in order of preference, we
consider the Roommates Problem (RP) and the Marriage Problem (MP) of matching
people with their partners. In RP there is no further restriction, while in MP
only people of opposite genders can be acceptable partners. For a pair of
matchings $X$ and $Y$, let $\phi(X,Y)$ denote the number of people who prefer a
person they get matched by $X$ to a person they get matched by $Y$, and define
an unpopularity factor $u(M)$ of a matching $M$ to be the maximum ratio
$\phi(M',M) / \phi(M,M')$ among all other possible matchings $M'$. In this
paper, we develop an algorithm to compute the unpopularity factor of a given
matching in $O(m\sqrt{n}\log^2 n)$ time for RP and in $O(m\sqrt{n}\log n)$ time
for MP, where $m$ is the total length of people's preference lists. We also
generalize the notion of unpopularity factor to a weighted setting where people
are given different voting weights and show that our algorithm can be slightly
modified to support that setting with the same running time.

31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys