Node Max-Cut and Computing Equilibria in Linear Weighted Congestion Games

TL;DR

This paper investigates the computational complexity of finding local optima and pure Nash equilibria in weighted congestion games and a related Max-Cut variant, revealing PLS-completeness results and efficient approximation methods.

Contribution

It proves PLS-completeness of PNE computation in certain weighted congestion games and introduces Node-Max-Cut, a restricted Max-Cut variant, with complexity and approximation results.

Findings

Computing PNE is PLS-complete for restricted congestion games.

Node-Max-Cut is PLS-complete even with vertex-weighted graphs.

A (1+ε)-approximate equilibrium can be efficiently computed when vertex weights are limited.

Abstract

In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkable gap regarding the complexity of PNE in congestion games with weighted and unweighted players, since in case of unweighted players, a PNE can be easily computed by either a simple greedy algorithm (for series-parallel networks) or any better response dynamics (when the latency is equal to the congestion). For the latter of the results above, we need to show first that computing a local optimum of a natural restriction of Max-Cut, which we call \emph{Node-Max-Cut}, is PLS-complete. In Node-Max-Cut, the input graph is vertex-weighted and the weight of each edge is equal to the product of the weights of its endpoints. Due to the very restricted nature of Node-Max-Cut, the reduction requires a careful combination of new gadgets with ideas and techniques from previous work. We also show how to compute efficiently a $(1+\eps)$-approximate equilibrium for Node-Max-Cut, if the number of different vertex weights is constant.