Multi-winner Approval Voting with Grouped Voters

We consider the general case of approval-based committee elections, where some attributes divide the voters into diverse groups which vary in size. This scenario occurs in applications like the presidential election, where voters come from different parties, or the student board election at a university with students from different schools. However, all existing committee election rules either are derived for the single-group case, or neglect the welfare of groups with few votes. Therefore, new voting rules are needed for this setting. In this paper, We propose two natural axioms for this setting, namely, small group benefited representation (SGBR) and large group benefited representation (LGBR). SGBR requires that if the committee size exceeds the number of groups, at least one candidate approved by each group is in the winning committee. LGBR requires that the winning committee must have at least as many candidates approved by a large group as by a small group. Based on the axioms, we propose three models and investigate parameterized complexity of the models with respect to various parameters. We show that all models are fixed-parameter tractable (FPT) when parameterized by the number n of votes, whereas they become fixed-parameter intractable when parameterized by the size k of the committee or d of the satisfaction bound.