Optimal versus Nash Equilibrium Computation for Networked Resource Allocation

TL;DR

This paper investigates the computational complexity of optimal and Nash equilibrium resource allocations in networked systems, providing efficient algorithms for certain cases and establishing NP-hardness for others, with implications for web caches and peer-to-peer networks.

Contribution

It introduces approximation algorithms for resource allocation and Nash equilibria, and proves NP-hardness results, linking Nash equilibria to Max-k-Cut solutions in networked resource problems.

Findings

Optimal allocation for 2 resources is efficiently computable.

NP-hardness of resource allocation for more than 2 resources.

A 3-approximation algorithm for resource allocation and Nash equilibria.

Abstract

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.