When Votes Change and Committees Should (Not)

TL;DR

This work introduces two time-aware multistage voting models for small-committee selection under Plurality: the conservative MSNTV and the revolutionary RMSNTV, grounding them in a sequence of voting profiles with constraints on committee size, stage-wise score, and changes between consecutive stages. The authors establish NP-hardness even for minimal agent counts, and provide a thorough parameterized complexity landscape, showing XP membership with W[1]-hardness in the number of stages and contrasting the two models (revolutionary often easier under certain parameter regimes). They develop kernelization and preprocessing results, including polynomial kernels when combining the number of stages with other parameters, and provide explicit constructions (e.g., in-out graphs, gadgets, and Sidon-set based reductions) to prove hardness. Collectively, the results delineate the computational boundaries of multistage committee selection and offer practical tractable cases for small parameters, supported by data-reduction techniques and structured algorithmic frameworks. The findings have implications for planning-oriented applications where time-varying preferences and stable or deliberately shifting committees must be balanced, and they open avenues for approximate, online, and broader voting-rule extensions.

Abstract

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce and study two time-dependent multistage models based on simple Plurality voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our task is to find a small committee for each stage of high score. In the conservative model we additionally require that any two consecutive committees have a small symmetric difference. Analogously, in the revolutionary model we require large symmetric differences. We prove both models to be NP-hard even for a constant number of agents, and, based on this, initiate a parameterized complexity analysis for the most natural parameters and combinations thereof. Among other results, we prove both models to be in XP yet W[1]-hard regarding the number of stages, and that being revolutionary seems to be "easier" than being conservative: If the (upper- resp. lower-) bound on the size of symmetric differences is constant, the conservative model remains NP-hard while the revolutionary model becomes polynomial-time solvable.

Figures (5)

• Figure 1: Overview of results for MSNTV and RMSNTV. Abbreviations p-$\operatorname{NP}$-h and $\operatorname{W[1]}$-h stand for, respectively, para-$\operatorname{NP}$-hard and $\operatorname{W[1]}$-hard. An arrow from one parameter $p$ to another parameter $p'$ indicates that $p$ can be upper bounded by some function in $p'$ (e.g., $\ell\leq 2k$, $k\leq m$, or $x\leq n$). The spiderweb diagram depicts further results being not displayed for readability (solid: conservative; dashed: revolutionary). a(\ref{['thm:xpregardingk', 'thm:cmpvwhardktau']}) b (\ref{['thm:xptau', 'thm:cmpvwhardktau']}) c (\ref{['thm:xptau', 'thm:rmpvwhardtau']}) $^\dagger$KRZ21
• Figure 2: Overview of results for MSNTV and RMSNTV. Abbreviations PK, noPK, p-$\operatorname{NP}$-h, and $\operatorname{W[1]}$-h stand for, respectively, "polynomial kernel", "no polynomial kernel unless $\operatorname{NP}\subseteq \operatorname{coNP}/\operatorname{poly}$", para-$\operatorname{NP}$-hard, and $\operatorname{W[1]}$-hard. An arrow from one parameter $p$ to another parameter $p'$ indicates that $p$ can be upper bounded by some function in $p'$ (e.g., $\ell\leq 2k$, $k\leq m$, or $x\leq n$). "?" indicates that a detailed classification is unknown. The spiderweb diagram depicts further results being not displayed for readability (solid: conservative; dashed: revolutionary). a(\ref{['thm:xpregardingk', 'thm:cmpvwhardktau']}) b (\ref{['thm:xptau', 'thm:cmpvwhardktau']}) c (\ref{['thm:xptau', 'thm:rmpvwhardtau']}) $^\dagger$KRZ21
• Figure 3: Illustration of the construction in the proof of \ref{['thm:cmpvwhardktau']}, exemplified with edge $e=\{v,w\}\in E_i^j$ with $v\in V_i$ and $w\in V_j$. A column represents a stage (which in turn represents an vertex or edge selection gadget for some vertex or edge set, respectively, or a coherence gadget of a pair of colors) and a row represents an agent (approving either a vertex or an edge). For brevity, we use $\mathop{\mathrm{id}}({v,w})$ to denote $\mathop{\mathrm{id}}({v}) + \mathop{\mathrm{id}}({w})$.
• Figure 4: Illustration of an in-out graph from \ref{['def:inoutgraph']}.
• Figure :

Theorems & Definitions (60)

• Theorem 1: \ref{['proof:thm:bothnphard']}
• Proposition 1
• proof
• Lemma 1: \ref{['proof:lem:cmpvtormpv']}
• proof
• Corollary 1
• Proposition 2
• Lemma 2
• proof
• Lemma 3