The fair allocation of mixed goods, consisting of both divisible and indivisible goods, has been a prominent topic of study in economics and computer science. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function. This fairness criterion includes standard ones such as maximum egalitarian social welfare and maximum Nash social welfare. We address the problem of minimizing a given symmetric strictly convex function when agents have binary valuations. If only divisible goods or only indivisible goods exist, the problem is known to be solvable in polynomial time. In this paper, firstly, we demonstrate that the problem is NP-hard even when all indivisible goods are identical. This NP-hardness is established even for maximizing egalitarian social welfare or Nash social welfare. Secondly, we provide a polynomial-time algorithm for the problem when all divisible goods are identical. To accomplish these, we exploit the proximity structure inherent in the problem. This provides theoretically important insights into the hybrid domain of convex optimization that incorporates both discrete and continuous aspects.

翻译：混合商品（包含可分割与不可分割商品）的公平分配问题一直是经济学与计算机科学领域的重要研究课题。我们将分配定义为公平的当且仅当其效用向量最小化一个对称严格凸函数。该公平性准则涵盖标准准则，如最大平等社会福利和最大纳什社会福利。本文研究当代理具有二元估值时最小化给定对称严格凸函数的问题。若仅存在可分割商品或仅存在不可分割商品，该问题已知可在多项式时间内求解。首先，我们证明即使所有不可分割商品完全相同，该问题仍为NP难问题——即便针对最大化平等社会福利或纳什社会福利的情形，此NP难性依然成立。其次，针对所有可分割商品完全相同的情形，我们提出多项式时间算法。为实现上述目标，我们利用了该问题固有的邻近结构。这为兼具离散与连续特征的凸优化混合领域提供了重要的理论洞见。