Complexity and Approximability of Social Welfare Optimization in Multiagent Resource Allocation

A central task in multiagent resource allocation, which provides mechanisms to allocate (bundles of) resources to agents, is to maximize social welfare. We assume resources to be indivisible and nonshareable and agents to express their utilities over bundles of resources, where utilities can be represented in the bundle form, the k-additive form, and as straight-line programs. We study the computational complexity of social welfare optimization in multiagent resource allocation, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product. We prove that exact social welfare optimization by the Nash product is DP-complete for the bundle and the 3-additive form, where DP is the second level of the boolean hierarchy over NP. For utility functions represented as straight-line programs, we show NP-completeness for egalitarian social welfare optimization and social welfare optimization by the Nash product. Finally, we show that social welfare optimization by the Nash product in the 1-additive form is hard to approximate, yet we also give fully polynomial-time approximation schemes for egalitarian and Nash product social welfare optimization in the 1-additive form with a fixed number of agents.