Group Control for Procedural Rules: Parameterized Complexity and Consecutive Domains
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We consider Group Control by Adding Individuals (GCAI) for two procedural rules – the consensus-start-respecting rule and the liberal-start-respecting rule. It is known that GCAI for both rules are NP-hard, but whether they are fixed-parameter tractable with respect to the number of distinguished candidates remained open. We resolve both open problems in the affirmative. In addition, we strengthen the NP-hardness of GCAI by showing that, with respect to the natural parameter the number of added individuals, GCAI for both rules is W[2]-hard. Notably, the W[2]-hardness for the liberal-start-respecting rule holds even when restricted to a very special case where the qualifications of individuals satisfy the so-called consecutive ones property. However, for the consensus-start-respecting rule, the problem becomes polynomial-time solvable in this special case. We also study a duality restriction where the disqualifications of individuals fulfill the consecutive ones property, and we show that under this restriction GCAI for both rules turn out to be polynomial-time solvable. Our reductions for showing W[2]-hardness also imply several other lower bounds concerning kernelization and exact algorithms.
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