Complexity and Approximability of Optimal Resource Allocation and Nash Equilibrium over Networks

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.