The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level, our model comprises a graph whose nodes represent the agents, which can be thought of as the sources of pollution in the network. The edges between agents represent the effect of spread of pollution. The government who is the regulator, is responsible for the maximization of the social welfare and sets bounds on the levels of emitted pollution in both local areas as well as globally in the whole network. We first prove that the above optimization problem is NP-hard even on some special cases of graphs such as trees. We then turn our attention on the classes of trees and planar graphs which model realistic scenarios of the emitted pollution in water and air, respectively. We derive approximation algorithms for these two kinds of networks and provide deterministic truthful and truthful in expectation mechanisms. In some settings of the problem that we study, we achieve the best possible approximation results under standard complexity theoretic assumptions. Our approximation algorithm on planar graphs is obtained by a novel decomposition technique to deal with constraints on vertices. We note that no known planar decomposition techniques can be used here and our technique can be of independent interest. For trees we design a two level dynamic programming approach to obtain an FPTAS. This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multi-dimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size. We furthermore derive truthful in expectation mechanisms on general networks with bounded degree.

中文翻译：

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网络污染游戏

污染控制问题主要在环境经济学文献中研究，其中博弈论的方法被应用于污染控制。据我们所知，这是第一次从计算的角度研究这个问题。我们引入了一种新的污染控制网络模型，并介绍了该模型的两个应用。在高层次上，我们的模型包含一个图，其节点代表代理，可以将其视为网络中的污染源。代理之间的边缘代表污染扩散的影响。作为监管者的政府负责实现社会福利的最大化，并对整个网络的本地和全球排放污染水平设定界限。我们首先证明上述优化问题是 NP-hard 的，即使在一些特殊的图（例如树）上也是如此。然后，我们将注意力转向树木和平面图的类别，它们分别模拟了水和空气中排放污染的现实场景。我们为这两种网络推导出近似算法，并在期望机制中提供确定性的真实性和真实性。在我们研究的问题的某些设置中，我们在标准复杂性理论假设下获得了可能的最佳近似结果。我们在平面图上的近似算法是通过一种新颖的分解技术来处理顶点约束的。我们注意到这里不能使用已知的平面分解技术，我们的技术可以是独立的。对于树，我们设计了一个两级动态规划方法来获得 FPTAS。这种方法对于应对全球污染配额限制至关重要。它使用一种特殊的多项选择、多维背包问题，其中除一个约束之外的所有约束的系数都受输入大小的多项式限制。我们进一步推导出具有有界度的一般网络的期望机制是真实的。