Chamberlin–Courant on Restricted Domains

The Chamberlin–Courant rule picks a committee so that every voter is represented by their most-preferred member, keeping overall dissatisfaction low. Finding such a committee is NP-hard in general, but polynomial-time solvable for single-peaked and single-crossing electorates. I look at how far that tractability actually stretches. For profiles a few candidates away from these domains, fixed-parameter tractable algorithms exist; for profiles a few voters away, XP algorithms. On robustness, deciding whether a small number of swaps in the votes can change the winning committee is NP-hard, even on nearly structured profiles. I also suggest a checklist for broadening domain restrictions further.