Over the past few years, the (parameterized) complexity landscape of constructive control for many prevalent approval-based multiwinner voting (ABMV) rules has been explored. We expand these results in two directions. First, we study constructive control for sequential Thiele's rules. Second, we study destructive counterparts of these problems. Our exploration leads to a comprehensive understanding of the complexity of these problems. Along the way, we also study several interesting axiomatic properties of ABMV rules, and obtain generic results for rules fulfilling these properties. In particular, we show that for many rules satisfying these properties, election control problems are generally hard to solve from a parameterized complexity point of view, even when restricted to certain special cases.

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