Optimizing Multiple Simultaneous Objectives for Voting and Facility Location

We study the classic facility location setting, where we are given n clients and m possible locations in some arbitrary metric space, want to choose a build facility. The exact same setting also arises spatial social choice, voters goal is candidate or outcome, with distance from voter an outcome representing cost of this for (e.g., based on their ideological differences). Unlike most previous work, do not focus single objective optimize total facility, maximum distance, etc.), but instead attempt several different objectives simultaneously. More specifically, consider l-centrum family objectives, which includes max many others. present tight bounds how well any pair such sum) can be simultaneously approximated compared optimum outcomes. In particular, show that it always approximates both within factor 1 plus square root 2, give precise characterization improves as two being optimized become more similar. For q>2 centrum approximate all q these small constant, constant approaches 3 increases. Our results when optimizing only few simultaneous form significantly better than approximation objectives.