  The winner determination problems of many attractive multi-winner voting
rules are NP-complete. However, they often admit polynomial-time algorithms
when restricting inputs to be single-peaked. Commonly, such algorithms employ
dynamic programming along the underlying axis. We introduce a new technique:
carefully chosen integer linear programming (IP) formulations for certain
voting problems admit an LP relaxation which is totally unimodular if
preferences are single-peaked, and which thus admits an integral optimal
solution. This technique gives efficient algorithms for finding optimal
committees under Proportional Approval Voting (PAV) and the Chamberlin--Courant
rule with single-peaked preferences, as well as for certain OWA-based rules.
For PAV, this is the first technique able to efficiently find an optimal
committee when preferences are single-peaked. An advantage of our approach is
that no special-purpose algorithm needs to be used to exploit structure in the
input preferences: any standard IP solver will terminate in the first iteration
if the input is single-peaked, and will continue to work otherwise.
