On the Parameterized Complexity of Controlling Amendment and Successive Winners

TL;DR

The paper addresses the parameterized complexity of eight standard election control problems under amendment, full-amendment, and successive procedures, and extends to the general $h$-amendment variant. It develops a broad landscape of fixed-parameter tractability and hardness results (including extsf{W[1]}-, extsf{W[2]}-hardness and paraNP-hardness) across multiple parameters, using reductions from problems such as RBDS, Perfect Code, Clique, and Biclique, as well as ILP-based extsf{FPT} techniques via extsf{MGCEV}. It reveals distinct computational behaviors between the amendment family and the successive procedure, with amendment generally more resistant to voter-control and successive more resistant to candidate-control, and shows that full-amendment and related variants add further complexity. The work also provides kernelization and approximation lower bounds, as well as several exact-algorithm insights, thereby delivering a thorough theoretical foundation for the resistance of these procedures to strategic manipulation and guiding future research directions, including experimental validation and broader control settings.

Abstract

The successive and the amendment procedures have been widely employed in parliamentary and legislative decision making and have undergone extensive study in the literature from various perspectives. However, investigating them through the lens of computational complexity theory has not been as thoroughly conducted as for many other prevalent voting procedures heretofore. To the best of our knowledge, there is only one paper which explores the complexity of several strategic voting problems under these two procedures, prior to our current work. To provide a better understanding of to what extent the two procedures resist strategic behavior, we study the parameterized complexity of constructive/destructive control by adding/deleting voters/candidates for both procedures. To enhance the generalizability of our results, we also examine a more generalized form of the amendment procedure. Our exploration yields a comprehensive (parameterized) complexity landscape of these problems with respect to numerous parameters.

Figures (9)

• Figure 1: The weighted majority graph of $(C, V)$ and the agenda as used in the proof of Theorem \ref{['thm-ccav-amd-np']}. All arcs among vertices in $R$ are forward arcs with a uniform weight of $\kappa+3$. The agenda is represented by the left-to-right ordering of the vertices.
• Figure 2: The weighted majority graph of $(C, V)$ and the agenda as used in the proof of Theorem \ref{['thm-ccdv-amd-np']}. All arcs among the vertices in $R$ are forward arcs with a weight of at least $|B| - \kappa + 1$. The agenda is represented by the left-to-right ordering of the vertices.
• Figure 3: An illustration of the weighted majority graph of $(C, V)$ and the agenda as used in the proof of Theorem \ref{['thm-dcav-amd-np']}. All arcs among the vertices in $X$ (respectively, $Y$) are forward arcs with a uniform weight of $3\kappa + 3$. The agenda is represented by the left-to-right ordering of the vertices.
• Figure 4: An illustration of the weighted majority graph of $(C, V)$ and the agenda as used in the proof of Theorem \ref{['thm-dcdv-amd-np']}. All arcs among the vertices in $R$ are forward with a weight of at least $2|B| - \kappa - \ell + 1$. The agenda is represented by the left-to-right ordering of the vertices.
• Figure 5: The sequence $(c_{11}, c_6, c_2)$ (highlighted as dark vertices) represents a $(c_{2} \leftarrow c_{11})$-beating path with respect to the subset $C' = \{c_1, c_2, c_4, c_5, c_6, c_7, c_{10}, c_{11}\}$ (vertices enclosed in blue circles), the election $(C', V)$, and the agenda $\rhd = (c_1, c_2, \dots, c_{12})$. A dark arc $(a, b)$ signifies that $a$ beats $b$, while a gray arc $(a, b)$ indicates that $a$ either beats or ties with $b$.

Theorems & Definitions (89)

• Definition 1: Parameterized Reduction
• Lemma 1
• proof
• Corollary 1: DBLP:journals/tcs/LiuFZL09DBLP:journals/tcs/LiuZ13
• Corollary 2: Bartholdi92howhard
• Corollary 3: Bartholdi92howhard
• Theorem 1
• proof
• Theorem 2
• proof