Yuhao Zhang proves exact MMS allocations always exist for goods and chores

Yuhao Zhang has resolved a significant open problem in algorithmic fair division. In a new arXiv paper titled "Exact MMS Allocations under Personalized Bivalued Valuations: Goods and Chores," the author proves that exact maximin share (MMS) allocations always exist for both indivisible goods and chores, even when each agent assigns one of two agent-specific values to every item. This "personalized bivalued" setting has been a key test case for fairness research, as prior attempts could only guarantee approximate allocations. The proof tackles the problem head-on with a novel combination of a quota-based reformulation and an envelope relaxation, followed by a sparse extreme-point construction and a flow-based rounding step that carefully controls total rounding loss.

Beyond the existential guarantee, Zhang's approach yields a polynomial-time algorithm to actually compute these allocations. This is a decisive step forward: previous work had shown that exact MMS allocations could be computed in some special cases, but the general bivalued case remained open. By settling it for both goods and chores, the result removes a major obstacle to practical fair division systems. The paper has been posted on arXiv (2608.15822) and is under the Computer Science and Game Theory (cs.GT) and Data Structures and Algorithms (cs.DS) categories. It directly addresses open questions highlighted by Ebadian, Peters, and Shah, as well as Garg, Huang, and Segal-Halevi. For anyone building resource allocation algorithms, this result offers a provable foundation for fair and efficient division in settings ranging from household chores to multi-agent systems.