On the Complexity of Candidates-Embedded Multiwinner Voting under the Hausdorff Function

Yongjie Yang

Abstract: We study candidates-embedded approval-based multiwinner voting. In this model, we are given a metric ƒ on the set of candidates, and voters are free to approve or disapprove any candidates. The task is to select a k-committee that either minimizes the sum of distances from the committee to all votes (utilitarian rules) or minimizes the maximum distance from the committee to any vote (egalitarian rules). The distance from a committee to a vote is measured by certain set-to-set functions derived from ƒ. Previous works have considered the min, the max, and the sum functions. This paper examines the Hausdorff function. We show that in general computing winners under the Hausdorff function is hard, but we also derive several polynomial-time algorithms for certain special cases.