Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

We study the existence of approximate pure Nash equilibria ( α -PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d . Previously, it was known that d -PNE always exist, whereas nonexistence was established only for small constants, namely, for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have Θ ˜ ( d ) -PNE, which provides the first superconstant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, by deploying this technique, we are able to show that deciding whether a weighted congestion game has an O ˜ ( d ) -PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest, and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of α -PNE, in which a certain set of players plays a specific strategy profile, is NP-hard for any α < 3 d / 2 , even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n . We show that n -PNE always exists, matched by an almost tight nonexistence bound of Θ ˜ ( n ) , which we can again transform into an NP-completeness proof for the decision problem.