Mathematical Programming formulations for the efficient solution of the $k$-sum approval voting problem

TL;DR

This paper develops mathematical programming models to efficiently solve the $k$-sum approval voting problem, enabling exact solutions for medium-sized elections with up to 200 voters and 60 candidates.

Contribution

It introduces novel mathematical formulations for the $k$-sum approval voting problem, facilitating exact solutions within practical computational times.

Findings

Efficient solution of medium-sized problems (up to 200 voters, 60 candidates).

Exact optimal solutions found quickly in all tested cases.

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.