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Complexity and Approximability of Optimal Resource Allocation and Nash Equilibrium over Networks
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-03-12 , DOI: 10.1137/19m1242525 S. Rasoul Etesami
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-03-12 , DOI: 10.1137/19m1242525 S. Rasoul Etesami
SIAM Journal on Optimization, Volume 30, Issue 1, Page 885-914, January 2020.
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, which can be viewed as a homogeneous data placement problem, 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 $k\ge 3$ this problem becomes NP-hard with no polynomial-time approximation algorithm with a performance guarantee better than $1+\frac{1}{102k^2}$, even under metric access costs. We then provide a $3$-approximation algorithm for the optimal resource allocation which runs only in $O(kn^2)$. 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. While this reduction suggests that finding a pure Nash equilibrium using best response dynamics might be PLS-hard, it allows us to use tools from complementary slackness and quadratic programming to devise systematic and more efficient algorithms towards obtaining Nash equilibrium points.
中文翻译：
网络上最优资源分配和Nash均衡的复杂性和逼近性
SIAM优化杂志，第30卷，第1期，第885-914页，2020年1月。
受新兴资源分配和数据放置问题（例如Web缓存和对等系统）的影响，我们考虑并研究了代理（节点）网络上的一类资源分配问题。在此模型中，可以将其视为同质数据放置问题，节点只能存储有限数量的资源，而通过其最近的邻居访问其余资源。我们在优化和博弈论框架下都考虑了这个问题。在最优资源分配的情况下，我们将首先表明，当只有$ k = 2 $资源时，可以在$ O（n ^ 2 \ log n）$个步骤中有效地找到最优分配，其中$ n $表示节点总数。然而，对于$ k \ ge 3 $，即使没有度量访问成本，该问题也变得难以解决，没有多项式时间近似算法，其性能保证优于$ 1 + \ frac {1} {102k ^ 2} $。然后，我们为仅在$ O（kn ^ 2）$中运行的最佳资源分配提供了一种$ 3 $近似算法。随后，我们在公式化为非合作博弈的自私环境下研究此问题，并提供了一种3美元的近似算法，用于在度量访问成本下获得其纯纳什均衡。然后，我们在特定加权完整图上的纯Nash均衡集与Max- $ k $ -Cut问题的翻转最优解之间建立了等价关系。虽然这种减少表明使用最佳响应动力学找到纯Nash平衡可能是PLS难以做到的，
更新日期：2020-03-12
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, which can be viewed as a homogeneous data placement problem, 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 $k\ge 3$ this problem becomes NP-hard with no polynomial-time approximation algorithm with a performance guarantee better than $1+\frac{1}{102k^2}$, even under metric access costs. We then provide a $3$-approximation algorithm for the optimal resource allocation which runs only in $O(kn^2)$. 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. While this reduction suggests that finding a pure Nash equilibrium using best response dynamics might be PLS-hard, it allows us to use tools from complementary slackness and quadratic programming to devise systematic and more efficient algorithms towards obtaining Nash equilibrium points.
中文翻译：
网络上最优资源分配和Nash均衡的复杂性和逼近性
SIAM优化杂志，第30卷，第1期，第885-914页，2020年1月。
受新兴资源分配和数据放置问题（例如Web缓存和对等系统）的影响，我们考虑并研究了代理（节点）网络上的一类资源分配问题。在此模型中，可以将其视为同质数据放置问题，节点只能存储有限数量的资源，而通过其最近的邻居访问其余资源。我们在优化和博弈论框架下都考虑了这个问题。在最优资源分配的情况下，我们将首先表明，当只有$ k = 2 $资源时，可以在$ O（n ^ 2 \ log n）$个步骤中有效地找到最优分配，其中$ n $表示节点总数。然而，对于$ k \ ge 3 $，即使没有度量访问成本，该问题也变得难以解决，没有多项式时间近似算法，其性能保证优于$ 1 + \ frac {1} {102k ^ 2} $。然后，我们为仅在$ O（kn ^ 2）$中运行的最佳资源分配提供了一种$ 3 $近似算法。随后，我们在公式化为非合作博弈的自私环境下研究此问题，并提供了一种3美元的近似算法，用于在度量访问成本下获得其纯纳什均衡。然后，我们在特定加权完整图上的纯Nash均衡集与Max- $ k $ -Cut问题的翻转最优解之间建立了等价关系。虽然这种减少表明使用最佳响应动力学找到纯Nash平衡可能是PLS难以做到的，
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