Selecting $k$ out of $m$ items based on the preferences of $n$ heterogeneous agents is a widely studied problem in algorithmic game theory. If agents have approval preferences over individual items and harmonic utility functions over bundles -- an agent receives $\sum_{j=1}^t\frac{1}{j}$ utility if $t$ of her approved items are selected -- then welfare optimisation is captured by a voting rule known as Proportional Approval Voting (PAV). PAV also satisfies demanding fairness axioms. However, finding a winning set of items under PAV is NP-hard. In search of a tractable method with strong fairness guarantees, a bounded local search version of PAV was proposed by Aziz et al. It proceeds by starting with an arbitrary size-$k$ set $W$ and, at each step, checking if there is a pair of candidates $a\in W$, $b\not\in W$ such that swapping $a$ and $b$ increases the total welfare by at least $\varepsilon$; if yes, it performs the swap. Aziz et al.~show that setting $\varepsilon=\frac{n}{k^2}$ ensures both the desired fairness guarantees and polynomial running time. However, they leave it open whether the algorithm converges in polynomial time if $\varepsilon$ is very small (in particular, if we do not stop until there are no welfare-improving swaps). We resolve this open question, by showing that if $\varepsilon$ can be arbitrarily small, the running time of this algorithm may be super-polynomial. Specifically, we prove a lower bound of~$\Omega(k^{\log k})$ if improvements are chosen lexicographically. To complement our lower bound, we provide an empirical comparison of two variants of local search -- better-response and best-response -- on several real-life data sets and a variety of synthetic data sets. Our experiments indicate that, empirically, better response exhibits faster running time than best response.

翻译：从$m$个候选项中选择$k$个，基于$n$个异质智能体的偏好，是算法博弈论中广泛研究的问题。若智能体对单个候选项具有批准偏好，且对候选包具有调和效用函数——当智能体批准的$t$个候选项被选中时，其获得$\sum_{j=1}^t\frac{1}{j}$的效用——则福利优化可通过称为比例批准投票（PAV）的投票规则实现。PAV同时满足严格的公平性公理。然而，在PAV下寻找获胜候选项集合是NP难的。为寻求具有强公平性保证的可行方法，Aziz等人提出了PAV的有界局部搜索版本。该算法从任意大小为$k$的集合$W$开始，在每一步检查是否存在候选对$a\in W$、$b\not\in W$，使得交换$a$和$b$能将总福利提升至少$\varepsilon$；若存在，则执行交换。Aziz等人证明设置$\varepsilon=\frac{n}{k^2}$可同时保证所需的公平性性质和多项式运行时间。然而，他们未解决当$\varepsilon$取值极小（特别是当算法持续执行直至不存在福利提升的交换操作时）算法是否能在多项式时间内收敛的问题。我们通过证明若$\varepsilon$可任意小，该算法的运行时间可能超多项式，从而解决了这一开放性问题。具体而言，我们证明了当改进按字典序选择时，算法具有$\Omega(k^{\log k})$的下界。为补充下界分析，我们在多个真实数据集和多种合成数据集上对局部搜索的两种变体——较优响应与最优响应——进行了实证比较。实验结果表明，较优响应在经验上展现出比最优响应更快的运行时间。