Minimizing Symmetric Convex Functions over a Hybrid of Continuous and Discrete Convex Sets
研究在积分基多面体与M-凸集的闵可夫斯基和上最小化对称严格凸函数的问题，该问题具有连续与离散混合结构，与混合商品公平分配相关，并证明了在特定条件下问题的NP难解性与多项式时间可解性。
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.