We consider Group Control by Adding Individuals (GCAI) in the setting of group identification 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 individuals 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 are 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 dual restriction where the disqualifications of individuals fulfill the consecutive ones property, and 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 lower bounds concerning kernelization and exact algorithms.

翻译：我们考虑在群体识别设定下，针对两种程序规则——共识起始尊重规则与自由起始尊重规则——的添加个体群体控制（GCAI）问题。已知这两种规则的GCAI问题均为NP难问题，但其关于受关注个体数量的固定参数可解性此前尚未解决。我们以肯定方式解决了这两个开放问题。此外，我们通过证明就自然参数“添加个体数”而言，两种规则的GCAI问题均为W[2]-难，加强了其NP难性。值得注意的是，自由起始尊重规则的W[2]-难性在个体资格满足所谓连续一性质这一特殊情形下仍然成立。然而在相同特殊情形下，共识起始尊重规则的问题变为多项式时间可解。我们还研究了对偶限制——个体不满足资格条件具有连续一性质，并证明在此限制下两种规则的GCAI问题均为多项式时间可解。我们的W[2]-难性归约还蕴含了关于核化与精确算法的若干下界结果。