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Computational Complexity of Manipulating Premise-Based Judgment Aggregation with Simple Formulas


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.
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Deeper Inquiries

What are some potential real-world applications of the manipulation problems studied in this paper

The manipulation problems studied in this paper have potential real-world applications in various decision-making scenarios where multiple opinions need to be aggregated. One application could be in political decision-making processes, where different stakeholders or political parties may try to manipulate the outcome by strategically influencing the aggregated judgments. Another application could be in financial decision-making, such as investment portfolio selection, where individuals or entities may try to manipulate the decision-making process to favor certain investments. Additionally, these manipulation problems could be relevant in online review systems, where businesses or individuals may try to manipulate the aggregated ratings to their advantage.

How can the insights from this paper be extended to other judgment aggregation rules beyond the uniform premise-based quota rule

The insights from this paper can be extended to other judgment aggregation rules beyond the uniform premise-based quota rule by adapting the problem formulations and computational complexity analyses to fit the specific rules. Different aggregation rules may have different properties and constraints, so the manipulation problems would need to be tailored to suit the specific characteristics of each rule. By applying similar methodologies and restrictions on the formulas, researchers can explore the computational complexity of manipulation in various judgment aggregation frameworks.

Are there other structural restrictions on the formulas, beyond the clause-based restrictions considered here, that could lead to tractable manipulation problems

Beyond the clause-based restrictions considered in this paper, there are other structural restrictions on the formulas that could lead to tractable manipulation problems. For example, restricting the formulas to be in disjunctive normal form (DNF) or conjunctive normal form (CNF) could provide a different set of tractable manipulation problems. Additionally, limiting the complexity of the logical connectives used in the formulas, such as only allowing conjunctions or disjunctions, could also impact the computational complexity of manipulation. Exploring these and other structural restrictions on the formulas could reveal new insights into the tractability of manipulation problems in judgment aggregation.
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