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We consider AC electrical systems where each electrical device has a powerdemand expressed as a complex number, and there is a limit on the magnitude oftotal power supply. Motivated by this scenario, we introduce the complex-demandknapsack 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 areal number often interpreted as weight or size of the item. While keeping thesame goal as to maximize the sum of values of the selected items, we put thecapacity limit on the magnitude of the sum of satisfied demands. For C-KP, weprove its inapproximability by FPTAS (unless P = NP), as well as presenting a(1/2-epsilon)-approximation algorithm. Furthermore, we investigate the selfishmulti-agent setting where each agent is in charge of one item, and an agent maymisreport the demand and value of his item for his own interest. We show asimple way to adapt our approximation algorithm to be monotone, which issufficient for the existence of incentive compatible payments such that noagent has an incentive to misreport. Our results shed insight on the design ofmulti-agent systems for smart grid.

Complex-Demand Knapsack Problems and Incentives in AC Power Systems