Judgment aggregation is a framework to aggregate individual opinions on multiple, logically connected issues into a collective outcome. It is open to manipulative attacks such as \textsc{Manipulation} where judges cast their judgments strategically. Previous works have shown that most computational problems corresponding to these manipulative attacks are \NP-hard. This desired computational barrier, however, often relies on formulas that are either of unbounded size or of complex structure. We revisit the computational complexity for various \textsc{Manipulation} and \textsc{Bribery} problems in judgment aggregation, now focusing on simple and realistic formulas. We restrict all formulas to be clauses that are (positive) monotone, Horn-clauses, or have bounded length. For basic variants of \textsc{Manipulation}, we show that these restrictions make several variants, which were in general known to be \NP-hard, polynomial-time solvable. Moreover, we 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 \textsc{Manipulation} and variants of \textsc{Satisfiability}. For Hamming distance based \textsc{Manipulation}, we show that \NP-hardness even holds for positive monotone clauses of length three, but the problem becomes polynomial-time solvable for positive monotone clauses of length two. For \textsc{Bribery}, we show that \NP-hardness even holds for positive monotone clauses of length two, but it becomes polynomial-time solvable for the same clause set if there is a constant budget.

翻译：判断聚合是一种将多个逻辑关联问题上的个体意见聚合为集体结果的框架。该过程容易受到操纵性攻击，例如\textsc{操纵}（Manipulation），其中判断者策略性地提交判断。以往研究表明，与这些操纵性攻击对应的多数计算问题均为\NP-困难。然而，这种期望的计算障碍通常依赖于无界规模或结构复杂的公式。我们重新审视判断聚合中各种\textsc{操纵}与\textsc{贿赂}（Bribery）问题的计算复杂性，重点关注简单且现实的公式。我们将所有公式限制为子句，这些子句是（正）单调的、Horn子句，或具有有界长度。对于\textsc{操纵}的基本变体，我们证明这些限制使得多个通常已知为\NP-困难的变体可在多项式时间内求解。此外，我们通过展示\textsc{操纵}变体与\textsc{可满足性}（Satisfiability）变体之间的紧密关系，提供了针对一大类子句限制（推广了单调子句和Horn子句）的P vs.\ \NP二分法。对于基于汉明距离的\textsc{操纵}，我们证明即使对于长度为三的正单调子句，\NP-困难性仍然成立，但对于长度为二的正单调子句，问题变为多项式时间可解。对于\textsc{贿赂}，我们证明即使对于长度为二的正单调子句，\NP-困难性仍然成立，但如果预算为常数，则对于同一子句集，问题变为多项式时间可解。