Maximum Nash Social Welfare under Budget-Feasible EFX

Fair division is a crucial problem in many research fields, such as computer science, economics, and cloud computing. In particular, the problem of allocating indivisible goods has been extensively studied, and the allocation satisfying the maximum Nash Social Welfare (NSW) is known to be envy-free up to one good (EF1), which is a widely-used fairness concept. However, EF1's fairness guarantee is limited to just one most valuable good, indicating a weakness in the scope. In this paper, we explore the relationship between maximum NSW and a stronger concept, envy-free up to any good (EFX), where envy disappears after removing any good. Our concentrate is on the budget-feasible setting, which ensures that every agent is allocated goods within her budget. When agents' values are in the Interval $[1, k]$ , any maximum NSW allocation ensures a $\frac{1}{k \cdot {(k-1)}}$ approximation of EFX, and in the Binary $\lbrace 0,1\rbrace$ case, it guarantees $\frac{1}{4}$ -EFX. Besides, we offer an algorithm for identifying a budget-feasible EFX allocation when agents have Binary valuations. Furthermore, we also improve the approximation ratio guarantee of budget-feasible EFX allocations by assuming the agents have large budgets.