Group Control for Procedural Rules: Parameterized Complexity and Consecutive Domains

TL;DR

This paper investigates the computational complexity of group control problems in group identification, establishing fixed-parameter tractability results and hardness bounds for two procedural rules under various structural restrictions.

Contribution

It resolves open questions on fixed-parameter tractability of GCAI for two rules and demonstrates W[2]-hardness, providing a comprehensive complexity analysis under different conditions.

Findings

GCAI is fixed-parameter tractable for both rules.

GCAI is W[2]-hard with respect to the number of added individuals.

Polynomial-time algorithms exist for certain special cases.

Abstract

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.