Participatory Budgeting (PB) is a form of participatory democracy in which citizens select a set of projects to be implemented, subject to a budget constraint. The Method of Equal Shares (MES), introduced in [18], is a simple iterative method for this task, which runs in polynomial time and satisfies a demanding proportionality axiom (Extended Justified Representation) in the setting of approval utilities. However, a downside of MES is that it is non-exhaustive: given an MES outcome, it may be possible to expand it by adding new projects without violating the budget constraint. To complete the outcome, the approach currently used in practice is as follows: given an instance with budget $b$, one searches for a budget $b'\ge b$ such that when MES is executed with budget $b'$, it produces a maximal feasible solution for $b$. The search is greedy, i.e., one has to execute MES from scratch for each value of $b'$. To avoid redundant computation, we introduce a variant of MES, which we call Adaptive Method of Equal Shares (AMES). Our method is budget-adaptive, in the sense that, given an outcome $W$ for a budget $b$ and a new budget $b'>b$, it can compute the outcome $W'$ for budget $b'$ by leveraging similarities between $W$ and $W'$. This eliminates the need to recompute solutions from scratch when increasing virtual budgets. Furthermore, AMES satisfies EJR in a certifiable way: given the output of our method, one can check in time $O(n\log n+mn)$ that it provides EJR (here, $n$ is the number of voters and $m$ is the number of projects). We evaluate the potential of AMES on real-world PB data, showing that small increases in budget typically require only minor modifications of the outcome.

翻译：参与式预算（PB）是一种公民在预算约束下选择待实施项目集的参与式民主形式。文献[18]提出的等额分配法（MES）是一种针对该任务的简单迭代方法，其运行时间为多项式复杂度，且在批准效用设置下满足严格的比例性公理（扩展正当代表性）。然而，MES的缺陷在于其非穷举性：给定MES输出结果后，可能通过添加新项目在预算约束内对其进行扩展。为获得完整结果，当前实践中采用的方法如下：给定预算$b$的实例，搜索满足预算$b'\ge b$使得使用$b'$执行MES时，能生成针对$b$的最大可行解。该搜索过程是贪婪的——即需要对每个$b'$值从头执行MES。为避免冗余计算，我们提出MES的变体——自适应等额分配法（AMES）。该方法具有预算自适应性：给定预算$b$的结果$W$和新预算$b'>b$时，可通过利用$W$与$W'$的相似性计算预算$b'$的结果$W'$。这消除了在增加虚拟预算时从头重新计算解的必要。此外，AMES以可验证方式满足EJR：通过我们方法的输出，可在$O(n\log n+mn)$时间内验证其是否满足EJR（其中$n$为选民数，$m$为项目数）。我们通过真实PB数据评估AMES的潜力，结果表明预算的小幅增长通常只需对结果进行微小调整。