2024/11/11 by Kurt Mehlhorn, Mehlhorn, Kurt
Engineering · #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #Scheduling and Optimization Algorithms
paper · pdf · doi:10.48550/arxiv.2411.06924
openalex publication_date 2024/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of m indivisible goods is to be allocated to a set of n agents. Each agent i has an additive valuation function vi over goods. The value of a good g for agent i is either 1 or s, where s is a fixed rational number greater than one, and the value of a bundle of goods is the sum of the values of the goods in the bundle. An allocation X is a partition of the goods into bundles X1, …, Xn, one for each agent. The Nash Social Welfare (\NSW) of an allocation X is defined as \NSW(X) = ( ∏i vi(Xi) )^\sfrac1n. The \NSW-allocation maximizes the Nash Social Welfare. In~\citeNSW-twovalues-halfinteger it was shown that the \NSW-allocation can be computed in polynomial time, if s is an integer or a half-integer, and that the problem is NP-complete otherwise. The proof for the half-integer case is quite involved. In this note we give a simpler and shorter proof