Complex-Demand Knapsack Problems and Incentives in AC Power Systems
  We consider AC electrical systems where each electrical device has a power demand expressed as a complex number, and there is a limit on the magnitude of total power supply. Motivated by this scenario, we introduce the complex-demand knapsack problem (C-KP), a new variation of the traditional knapsack problem, where each item is associated with a demand as a complex number, rather than a real number often interpreted as weight or size of the item. While keeping the same goal as to maximize the sum of values of the selected items, we put the capacity limit on the magnitude of the sum of satisfied demands. For C-KP, we prove its inapproximability by FPTAS (unless P = NP), as well as presenting a (1/2-epsilon)-approximation algorithm. Furthermore, we investigate the selfish multi-agent setting where each agent is in charge of one item, and an agent may misreport the demand and value of his item for his own interest. We show a simple way to adapt our approximation algorithm to be monotone, which is sufficient for the existence of incentive compatible payments such that no agent has an incentive to misreport. Our results shed insight on the design of multi-agent systems for smart grid. 
Lan Yu and Chi-Kin Chau
cs.DS