Motivated by emerging resource allocation and data placement problems such as web caches and peer-to-peer systems, we consider and study a class of resource allocation problems over a network of agents (nodes). In this model, nodes can store only a limited number of resources while accessing the remaining ones through their closest neighbors. We consider this problem under both optimization and game-theoretic frameworks. In the case of optimal resource allocation we will first show that when there are only k=2 resources, the optimal allocation can be found efficiently in O(n^2\log n) steps, where n denotes the total number of nodes. However, for k>2 this problem becomes NP-hard with no polynomial time approximation algorithm with a performance guarantee better than 1+1/102k^2, even under metric access costs. We then provide a 3-approximation algorithm for the optimal resource allocation which runs only in linear time O(n). Subsequently, we look at this problem under a selfish setting formulated as a noncooperative game and provide a 3-approximation algorithm for obtaining its pure Nash equilibria under metric access costs. We then establish an equivalence between the set of pure Nash equilibria and flip-optimal solutions of the Max-k-Cut problem over a specific weighted complete graph. Using this reduction, we show that finding the lexicographically smallest Nash equilibrium for k> 2 is NP-hard, and provide an algorithm to find it in O(n^3 2^n) steps. While the reduction to weighted Max-k-Cut suggests that finding a pure Nash equilibrium using best response dynamics might be PLS-hard, it allows us to use tools from quadratic programming to devise more systematic algorithms towards obtaining Nash equilibrium points.

中文翻译：

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网络资源分配的最优与纳什均衡计算

受网络缓存和点对点系统等新兴资源分配和数据放置问题的启发，我们考虑并研究了代理（节点）网络上的一类资源分配问题。在这个模型中，节点只能存储有限数量的资源，同时通过它们最近的邻居访问剩余的资源。我们在优化和博弈论框架下考虑这个问题。在最优资源分配的情况下，我们将首先证明当只有 k=2 个资源时，可以在 O(n^2\log n) 步中高效地找到最优分配，其中 n 表示节点总数。然而，对于 k>2，这个问题变得 NP-hard，没有多项式时间近似算法，性能保证优于 1+1/102k^2，即使在度量访问成本下也是如此。然后，我们为优化资源分配提供了一个 3 近似算法，该算法仅在线性时间 O(n) 内运行。随后，我们在制定为非合作博弈的自私设置下研究这个问题，并提供一种 3 近似算法，用于在度量访问成本下获得其纯纳什均衡。然后，我们在特定加权完全图上建立纯纳什均衡集和 Max-k-Cut 问题的翻转最优解之间的等价关系。使用这种减少，我们表明找到 k> 2 的字典序最小的纳什均衡是 NP 难的，并提供了一种算法来在 O(n^3 2^n) 步中找到它。虽然减少到加权 Max-k-Cut 表明使用最佳响应动力学找到纯纳什均衡可能是 PLS 困难的，