  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.
