The computational complexity of the bipartite popular matching problem depends on how indifference appears in the preference lists.
If one side has strict preferences while nodes on the other side are all indifferent (but prefer to be matched),
then a popular matching can be found in polynomial time [Cseh, Huang, Kavitha, 2015]. However, as the same paper points out, the
problem becomes NP-complete if nodes with strict preferences are allowed on both sides and indifferent
nodes are allowed on one side. We show that the problem of finding a \emph{strongly popular} matching
is polynomial-time solvable even in the latter case. More generally, we give a polynomial-time algorithm for the many-to-many version, i.e.\
the strongly popular $b$-matching problem, in the setting where one side has
strict preferences while agents on the other side may have one tie of arbitrary length at the end of their preference list.
Previous version can be found here.
Bibtex entry:
@techreport{egres-17-04,
AUTHOR | = | {Kir{\'a}ly, Tam{\'a}s and M{\'e}sz{\'a}ros-Karkus, Zsuzsa}, |
TITLE | = | {Finding strongly popular b-matchings in bipartite graphs}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2017}, |
NUMBER | = | {TR-2017-04} |