图上的公平、高效且连通的分配问题
Fair, Efficient and Connected Allocations on Graphs
浏览论文内容
中文总结 AI 辅助
该文研究图上高效连通分配问题的经典与参数化复杂度，在多参数下分析问题的可处理性与难解性，为该领域的算法设计提供了复杂度边界依据。
中文摘要 AI 辅助
我们研究图上高效连通分配问题的经典与参数化复杂度，其中效率通过平等主义与功利主义社会福利最大化来衡量。首先，我们在经典设定中确立了严格的复杂度二分：两类问题在一般图上均为NP-难，即使在路径、树、环等极受限图类上仍保持难解；而在星型图上可多项式时间求解，但该易处理性甚至无法扩展至两个不相交星型图的情况。受此边界启发，我们转向参数化复杂度框架，以智能体数量为参数研究问题，在树图上获得固定参数可处理性（FPT），并更广泛地发现连通图类的易处理性可扩展至该类图的不相交并集这一稳健现象。我们进一步研究树宽与树深参数，证明功利主义版本对二者均为FPT，而平等主义版本即使在树深为2的图上仍为para-NP-难。最后，我们分析连通分量数量，表明除星型图集合外，问题仍保持难解；对于星型图集合，平等主义情况为para-NP-难，而功利主义情况为W[2]-难且存在XP算法。
英文摘要
We study the classical and parameterized complexity of efficient connected allocation problems on graphs, where efficiency is measured by egalitarian and utilitarian welfare maximization. We first establish a sharp complexity dichotomy in the classical setting: both problems are NP-hard in general and remain hard even on very restricted graph classes such as paths, and consequently trees and cycles. In contrast, they are polynomial-time solvable on stars, but this tractability does not extend even to the case of two disjoint stars. Motivated by these boundaries, we move to the parameterized complexity framework, where we study the problem with respect to the number of agents. We obtain fixed-parameter tractability (FPT) on trees and, more generally, identify a robust phenomenon whereby tractability on a connected graph class extends to disjoint unions of graphs from that class. We further investigate the parameters treewidth and treedepth, showing that the utilitarian version is FPT for both, whereas the egalitarian version remains para-NP-hard even on graphs of treedepth two. Finally, we analyze the number of connected components and show that except for the collection of stars, the problems remain hard. For the collection of stars,while we obtain para-NP-hardness for the egalitarian case, the utilitarian case gives W[2]-hardness together with an XP algorithm.
