Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set indivisible goods among agents with subadditive valuations in fair and efficient manner. Envy-Freeness up to any good (EFX) is most compelling notion fairness context goods. Although existence EFX not known beyond simple case two valuations, some approximations are exist, namely 1/2-EFX allocation allocations bounded charity. Nash welfare (the geometric mean agents' valuations) one commonly used measures efficiency. In additive an that maximizes also satisfies properties like Envy-Free (EF1). there substantial work on approximating when have very little valuations. this paper, we design polynomial-time algorithm outputs either as well achieves O(n) approximation welfare. Our result improves current best-known O(n log n) O(m) submodular respectively. Furthermore, our technique gives family measures, p-mean for p (-\infty, 1], thereby matching asymptotically best ratio special cases = -\infty while retaining remarkable properties.