Impartial selection problems are concerned with the selection of one or more agents from a set based on mutual nominations from within the set. To avoid strategic nominations of the agents, the axiom of impartiality requires that the selection of each agent is independent of the nominations cast by that agent. This paper initiates the study of impartial selection problems where the nominations are weighted and the set of agents that can be selected is restricted by a combinatorial constraint. We call a selection mechanism $\alpha$-optimal if, for every instance, the ratio between the total sum of weighted nominations of the selected set and that of the best feasible set of agents is at least $\alpha$. We show that a natural extension of a mechanism studied for the selection of a single agent remains impartial and $\frac{1}{4}$-optimal for general independence systems, and we generalize upper bounds from the selection of multiple agents by parameterizing them by the girth of the independence system. We then focus on independence systems defined by knapsack and matroid constraints, giving impartial mechanisms that exploit a greedy order of the agents and achieve approximation ratios of $\frac{1}{3}$ and $\frac{1}{2}$, respectively, when agents cast a single nomination. For graphic matroids, we further devise an impartial and $\frac{1}{3}$-optimal mechanism for an arbitrary number of unweighted nominations.

翻译：公正选择问题关注如何基于集合内成员的相互提名，从集合中选出一个或多个个体。为避免个体策略性提名，公正性公理要求每个个体的选择独立于该个体自身的提名。本文首次研究了提名具有权重且可选个体集合受组合约束限制的公正选择问题。若对于任意实例，所选集合的加权提名总和与最优可行个体集合的加权提名总和之比至少为α，则称该选择机制为α-最优。我们证明，针对单个个体选择所研究的机制经自然扩展后，在一般独立系统中仍保持公正性且达到1/4最优性，并通过以独立系统的围长作为参数，推广了多个体选择的上界结果。随后我们聚焦于背包约束和拟阵约束定义的独立系统，提出利用个体贪心排序的公正机制，在个体仅进行单次提名时分别获得1/3和1/2的近似比。针对图拟阵，我们进一步设计了适用于任意数量未加权提名的公正且1/3最优机制。