Core Concepts
Restricting the conclusions to simple clauses can make several variants of manipulation problems, which were previously known to be NP-hard, polynomial-time solvable. However, NP-hardness still holds for some basic cases of manipulation even with simple formulas.
Abstract
The paper analyzes the computational complexity of various manipulation problems in premise-based judgment aggregation, focusing on the case where the conclusions are restricted to simple clauses.
Key highlights:
For basic variants of Manipulation (UPQR-Robustness-Manipulation and UPQR-Possible-Manipulation), the authors show that these problems become linear-time solvable when the conclusions are just simple clauses.
For UPQR-Necessary-Manipulation and UPQR-Exact-Manipulation, the authors show that these problems can be solved by solving at most |Φc| instances of a related Satisfiability problem, where Φc is the set of conclusions.
The authors provide a P vs. NP dichotomy for a large class of clause restrictions (generalizing monotone and Horn clauses) by showing a close relationship between variants of Manipulation and variants of Satisfiability.
For Hamming distance based Manipulation, the authors show that NP-hardness holds even for positive monotone clauses of length 3, but the problem becomes polynomial-time solvable for positive monotone clauses of length 2.
For Bribery, the authors show that NP-hardness holds even for positive monotone clauses of length 2, but it becomes polynomial-time solvable for the same clause set if there is a constant budget.