Truthful Mechanisms for Combinatorial Allocation of Electric Power in Alternating Current Electric Systems for Smart Grid release_nnpxsicitvdplmijt6csmsicj4

Abstract

Traditional studies of combinatorial auctions often only consider linear constraints. The rise of smart grid presents a new class of auctions, characterized by quadratic constraints. This paper studies 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 alternating current (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 ϕ∈[0,π/2-δ], and a truthful FPTAS, which fully optimizes the objective function but violates the capacity constraint by at most (1+ϵ), for the case ϕ∈(π/2,π-δ], where ϕ is the maximum argument of any complex-valued demand and ϵ,δ>0 are arbitrarily small constants. We complement these results by showing that, unless P=NP, neither a PTAS for the case ϕ∈(π/2,π-δ] nor any bi-criteria approximation algorithm with polynomial guarantees for the case when ϕ is arbitrarily close to π (that is, when δ is arbitrarily close to 0) can exist.
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