Abstract Several methods exist for making collective decisions on a set of variables when voters possibly have preferential dependencies. None is based on approval voting. We define a family of rules for approval-based voting on combinatorial domains, where voters cast conditional approval ballots, allowing them to approve values of a variable conditionally on the values of other variables. We study three such rules. The first two generalize simple multiwinner approval voting and minimax approval voting. The third one is an approval-based version of sequential voting on combinatorial domains. We study some properties of these rules, and compare their outcomes.