We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuation functions. Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods, while Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely recognized efficiency criteria. Although each property is straightforward to achieve individually, simultaneously ensuring both fairness and efficiency is challenging. Caragiannis et al.~\cite{caragiannis2019unreasonable} established the surprising result that maximizing Nash social welfare yields an allocation that is both EF1 and PO; however, since maximizing Nash social welfare is NP-hard, this approach does not provide an efficient algorithm. To overcome this barrier, Barman, Krishnamurthy, and Vaish~\cite{barman2018finding} designed a pseudo-polynomial time algorithm to compute an EF1 and PO allocation, and showed the existence of EF1 and fPO allocations. Nevertheless, the latter existence proof relies on a non-constructive convergence argument and does not directly yield an efficient algorithm for finding EF1 and fPO allocations. Whether a polynomial-time algorithm exists for finding an EF1 and PO (or fPO) allocation remains an important open problem. In this paper, we propose a polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed. Our primary idea is to avoid processing the entire instance at once; instead, we sequentially add agents to the instance and construct an allocation that satisfies EF1 and fPO at each step.

翻译：我们研究了在具有加性估值函数的代理之间公平高效地分配不可分割物品的问题。对于不可分割物品，**单物品无嫉妒性**（EF1）是一个被广泛研究的公平性概念，而**帕累托最优性**（PO）及其更强的变体**分数帕累托最优性**（fPO）则是公认的效率标准。尽管单独实现每个属性都相对直接，但同时确保公平与效率却具有挑战性。Caragiannis等人~\cite{caragiannis2019unreasonable} 建立了一个令人惊讶的结果：最大化纳什社会福利可以产生一个同时满足EF1和PO的分配；然而，由于最大化纳什社会福利是NP难问题，这种方法并未提供一个高效的算法。为了克服这一障碍，Barman、Krishnamurthy和Vaish~\cite{barman2018finding} 设计了一种伪多项式时间算法来计算EF1和PO分配，并证明了EF1和fPO分配的存在性。然而，后者的存在性证明依赖于一个非构造性的收敛论证，并未直接给出寻找EF1和fPO分配的高效算法。是否存在一种多项式时间算法来寻找EF1和PO（或fPO）分配，仍然是一个重要的开放性问题。在本文中，我们提出了一种多项式时间算法，用于在代理数量固定且估值函数为加性的情况下，计算一个同时满足EF1和fPO的分配。我们的核心思想是避免一次性处理整个实例；相反，我们依次向实例中添加代理，并在每一步都构建一个满足EF1和fPO的分配。