We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of $k$-ary instances where $k$ is a constant, i.e., each agent has at most $k$ different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.

翻译：本文研究在可加性估值下，使用流行的公平性概念——至多一件物品的无嫉妒性（EF1）和至多一件物品的公平性（EQ1），结合帕累托最优性（PO），对一组不可分割物品进行公平且高效分配的问题。已知存在伪多项式时间算法计算EF1+PO分配，以及存在既满足EF1又满足分数帕累托最优性（fPO，一种比PO更强的概念）分配的非构造性证明。我们提出一种计算EF1+fPO分配的伪多项式时间算法，从而改进了先前的结果。我们的技术还表明，当物品价值为正时，EQ1+fPO分配始终存在，且可在伪多项式时间内计算得出。我们还考虑一类$k$元实例（其中$k$为常数），即每个代理人对物品至多有$k$种不同估值。对于此类实例，我们证明可在强多项式时间内计算出EF1+fPO分配。当所有价值为正时，我们证明此类实例的EQ1+fPO分配可在强多项式时间内计算得出。接下来，我们考虑代理人数量为常数的实例，并证明可在多项式时间内计算出EF1+PO（同样适用于EQ1+PO）分配。这些结果显著扩展了多项式时间可计算性，使其超越已知的二元或相同估值情形。我们还设计了一个多项式时间算法，当代理人数为常数且每个代理人对物品的估值种类为常数时，该算法可计算纳什福利最大化分配。最后，在复杂性方面，我们证明计算EF1+fPO分配的问题属于PLS复杂性类。