Title: Mathematical Programming formulations for the efficient solution of the $k$-sum approval voting problem,
Abstract: In this paper we address the problem of electing a committee among a set of
$m$ candidates and on the basis of the preferences of a set of $n$ voters. We
consider the approval voting method in which each voter can approve as many
candidates as she/he likes by expressing a preference profile (boolean
$m$-vector). In order to elect a committee, a voting rule must be established
to `transform' the $n$ voters' profiles into a winning committee. The problem
is widely studied in voting theory; for a variety of voting rules the problem
was shown to be computationally difficult and approximation algorithms and
heuristic techniques were proposed in the literature. In this paper we follow
an Ordered Weighted Averaging approach and study the $k$-sum approval voting
(optimization) problem in the general case $1 \leq k <n$. For this problem we
provide different mathematical programming formulations that allow us to solve
it in an exact solution framework. We provide computational results showing
that our approach is efficient for medium-size test problems ($n$ up to 200,
$m$ up to 60) since in all tested cases it was able to find the exact optimal
solution in very short computational times.