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

• 成果类型: Article; Early Access
• 署名作者: Kawase, Yasushi; Nishimura, Koichi; Sumita, Hanna
• 署名单位: University of Tokyo
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2024.0659
• 发表日期: 2026-02-26
• 关键词: integral base-polyhedron; Fair allocation; matroid; maximum Nash welfare; strongly polynomial algorithm; NASH SOCIAL-WELFARE; base; fair

摘要

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 basepolyhedra 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.