Electing a committee with dominance constraints

Abstract

We consider the problem of electing a committee of k candidates, subject to some constraints as to what this committee is supposed to look like. In our framework, the candidates are given labels as an abstraction of gender, religion, ethnicity, and other attributes, and the election outcome is constrained by interval constraints -of the form "Between 3 and 5 candidates with label X" -and dominance constraints -"At least as many candidates with label X as with label Y". While in general this problem would require us to rethink how we determine which election outcomes are good, in the case of a committee scoring rule this becomes a constrained optimisation problem -simply find a valid committee with the highest score. In the case of weakly separable rules we show the existence of a polynomial time solution in the case of tree-like constraints, and a fixed-parameter tractable algorithm for the general case, which is otherwise NP-hard.