We address the problem of configuring a power distribution network with reliability and resilience objectives by satisfying the demands of the consumers and saturating each production source as little as possible. We consider power distribution networks containing source nodes producing electricity, nodes representing electricity consumers and switches between them. Configuring this network consists in deciding the orientation of the links between the nodes of the network. The electric flow is a direct consequence of the chosen configuration and can be computed in polynomial time. It is valid if it satisfies the demand of each consumer and capacity constraints on the network. In such a case, we study the problem of determining a feasible solution that balances the loads of the sources, that is their production rates. We use three metrics to measure the quality of a solution: minimizing the maximum load, maximizing the minimum load and minimizing the difference of the maximum and the minimum loads. This defines optimization problems called respectively min-M, max-m and min-R. In the case where the graph of the network is a tree, it is known that the problem of building a valid configuration is polynomial. We show the three optimization variants have distinct properties regarding the theoretical complexity and the approximability. Particularly, we show that min-M is polynomial, that max-m is NP-Hard but belongs to the class FPTAS and that min-R is NP-Hard, cannot 1 be approximated to within any exponential relative ratio but, for any $\epsilon$ > 0, there exists an algorithm for which the value of the returned solution equals the value of an optimal solution shifted by at most $\epsilon$.

翻译：我们以可靠性和弹性为目标，研究了电力分配网络的配置问题，要求在满足用户需求的同时尽可能降低各生产源的负载饱和程度。考虑包含发电源节点、电力消费节点及连接开关的配电网。该网络的配置过程在于确定节点间链路的方向。电力流量直接由所选配置决定，且可在多项式时间内计算得到。若该配置能同时满足各用户的电力需求及网络容量约束，则视为有效配置。在此前提下，我们研究如何确定可行的解决方案以平衡各能源负载（即其发电速率）。采用三种指标衡量解的质量：最小化最大负载、最大化最小负载、以及最小化最大与最小负载之差，分别对应优化问题min-M、max-m与min-R。已知当网络拓扑为树形结构时，构建有效配置的问题可在多项式时间内求解。我们证明这三个优化变体在理论复杂度与近似性方面具有不同特性：特别地，min-M为多项式可解问题，max-m属于NP难问题但存在完全多项式时间近似方案(FPTAS)，而min-R为NP难问题且无法以任意指数相对比率进行近似，但对于任意ε>0，存在算法可使返回解的值与最优解的值相差不超过ε。