Minimizing Symmetric Convex Functions over a Hybrid of Continuous and Discrete Convex Sets

We study the problem of minimizing a given symmetric strictly convex function over the Minkowski sum of an integral base-polyhedron and an M-convex set. This problem has a hybrid of continuous and discrete structures. This relates to allocating mixed goods, consisting of both divisible and indivisible goods, to agents with binary valuations so that the fairness measure, such as the Nash welfare, is maximized. Integral base-polyhedra and M-convex sets have similar and nice properties, and the nonhybrid case can be solved in polynomial time. Whereas the hybrid case lacks some of these properties, we show structures of an optimal solution. Through our findings, we demonstrate that our problem is NP-hard even in the fair allocation setting where all indivisible goods are identical. Moreover, we provide a polynomial-time algorithm for the fair allocation problem when all divisible goods are identical. Funding: This work was supported by Precursory Research for Embryonic Science and Technology (JST PRESTO) [Grant JPMJPR2122], Exploratory Research for Advanced Technology (JST ERATO) [Grant JPMJER2301], the Japan Society for the Promotion of Science (JSPS KAKENHI) [Grants JP20K19739, JP21H03397, JP21K17708, and JP25K00137], and Value Exchange Engineering, a joint research project between R4D, Mercari, Inc., and the RIISE.