Approximability and Inapproximability of Social Welfare Optimization in Multiagent Resource Allocation

Resource allocation is a fundamental problem in the field of multiagent systems, where a set of resources need to be allocated to a group of agents in a way so as to optimizing social welfare. In the last few years this problem has received much attention, especially due to its wide applicability domain (see the survey of Chevaleyre et al. [CDE+06]). This thesis studies the approximability and inapproximability of several important optimization problems addressed in the area of multiagent resource allocation, focusing on two central representations of preferences, the bundle form and the k-additive form, and on various types of social welfare, ranging from quantitative measures (utilitarian, egalitarian and Nash product social welfare) to qualitative measures (envy-freeness). First of all, we study the social welfare maximization problems, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product (for both total and average versions). We obtain several approximability and inapproximability results. For the utilitarian social welfare maximization with 2-additive utilities, we show that unless P = NP, it is impossible to achieve an approximation factor better than 21/22. In addition, we prove that both egalitarian and Nash product social welfare maximization are NP-hard to approximate to within any factor, each for both the bundle form and the 3-additive form. For utility functions represented as additive form, we establish a hardness factor of (8/9) + ε for total Nash product social welfare maximization and another hardness factor of (2 √ 2/3) + ε for the average version. Our positive results have all been achieved regarding the additive form. We provide a fast greedy approximation algorithm to within a factor of (1/m−n+1) for average Nash social welfare maximization, where n and m are the number of agents and the number of resources, respectively. We also prove that this problem admits a PTAS for the case of identical agents. Particularly, we obtain an FPTAS for maximization problems with respect to egalitarian and Nash product social welfare when the number of agents is not the part of input. Finally, we consider a special case when n = m and present a polynomial-time algorithm that produces optimal solutions for both total and average Nash social welfare problems.