Approval-based committee (ABC) voting represents a well-studied multiwinner election setting, where a subset of candidates of a predetermined size, a so-called committee, needs to be chosen based on the approval preferences of a set of voters [23]. Traditionally, ABC voting is studied in the context where we know, for each voter and each candidate, whether the voter approves the candidate or not. In this paper, we investigate the situation where the approval information is incomplete. Specifically, we assume that each voter is associated with a set of approved candidates, a set of disapproved candidates, and a set of candidates where the voter's stand is unknown, hereafter referred to as the unknown candidates. Moreover, we may have (partial) ordinal information on voters' preferences among the unknown candidates, restricting the "valid" completions of voters' approval sets. When the number of candidates is large, unknown candidates are likely to exist because voters are not aware of or not familiar with, and therefore cannot evaluate, all candidates. In particular, this holds in scenarios where candidates join the election over time, and voter preferences over new candidates have not been elicited [16].

We study two-stage committee elections where voters have dynamic preferences over candidates; at each stage, a committee is chosen under a given voting rule. We are interested in identifying a winning committee for the second stage that overlaps as much as possible with the first-stage committee. We show a full complexity dichotomy for the class of Thiele rules: this problem is tractable for Approval Voting (AV) and hard for all other Thiele rules (including, in particular, Proportional Approval Voting and the Chamberlin-Courant rule). We extend this dichotomy to the greedy variants of Thiele rules. We also explore this problem from a parameterized complexity perspective for several natural parameters. We complement the theory with experimental analysis: e.g., we investigate the average number of changes in the committee as a function of changes in voters' preferences and the role of ties.

We establish a link between multiwinner elections and apportionment problems by showing how approval-based multiwinner election rules can be interpreted as methods of apportionment. We consider several multiwinner rules and observe that they induce apportionment methods that are well-established in the literature on proportional representation. For instance, we show that Proportional Approval Voting induces the D'Hondt method and that Monroe's rule induces the largest reminder method. We also consider properties of apportionment methods and exhibit multiwinner rules that induce apportionment methods satisfying these properties.