Truthful Mechanisms for Combinatorial AC Electric Power Allocation
  Traditional studies of combinatorial auctions often only consider linear constraints (by which the demands for certain goods are limited by the corresponding supplies). The rise of smart grid presents a new class of auctions, characterized by quadratic constraints. Yu and Chau [AAMAS 13'] introduced the complex-demand knapsack problem, in which the demands are complex-valued and the capacity of supplies is described by the magnitude of total complex-valued demand. This naturally captures the power constraints in AC electric systems. In this paper, we provide a more complete study and generalize the problem to the multi-minded version, beyond the previously known 1/2-approximation algorithm for only a subclass of the problem. More precisely, we give a truthful PTAS for the case phi in [0,pi/2-delta], and a truthful FPTAS, which fully optimizes the objective function but violates the capacity constraint by at most (1+epsilon), for the case phi in (pi/2,pi-delta], where phi is the maximum angle between any two complex-valued demands and epsilon,delta>0 are arbitrarily small constants. 
Chi-Kin Chau, Khaled Elbassioni, Majid Khonji
cs.GT cs.DS