  We study the complexity of winner determination in single-crossing elections
under two classic fully proportional representation
rules---Chamberlin--Courant's rule and Monroe's rule. Winner determination for
these rules is known to be NP-hard for unrestricted preferences. We show that
for single-crossing preferences this problem admits a polynomial-time algorithm
for Chamberlin--Courant's rule, but remains NP-hard for Monroe's rule. Our
algorithm for Chamberlin--Courant's rule can be modified to work for elections
with bounded single-crossing width. To circumvent the hardness result for
Monroe's rule, we consider single-crossing elections that satisfy an additional
constraint, namely, ones where each candidate is ranked first by at least one
voter (such elections are called narcissistic). For single-crossing
narcissistic elections, we provide an efficient algorithm for the egalitarian
version of Monroe's rule.
