Pure Nash Equilibria in Graphical Games and Treewidth

We treat PNE-GG, the problem of deciding the existence of a Pure Nash Equilibrium in a graphical game, and the role of treewidth in this problem. PNE-GG is known to be (Formula presented.)-complete in general, but polynomially solvable for graphical games of bounded treewidth. We prove that PNE-GG is (Formula presented.)-Hard when parameterized by treewidth. On the other hand, we give a dynamic programming approach that solves the problem in (Formula presented.) time, where (Formula presented.) is the cardinality of the largest strategy set and (Formula presented.) is the treewidth of the input graph (and (Formula presented.) hides polynomial factors). This proves that PNE-GG is in (Formula presented.) for the combined parameter (Formula presented.). Moreover, we prove that there is no algorithm that solves PNE-GG in (Formula presented.) time for any (Formula presented.), unless the Strong Exponential Time Hypothesis fails. Our lower bounds implicitly assume that (Formula presented.); we show that for (Formula presented.) the problem can be solved in polynomial time. Finally, we discuss the implication for computing pure Nash equilibria in graphical games (PNE-GG) of (Formula presented.) treewidth, the existence of polynomial kernels for PNE-GG parameterized by treewidth, and the construction of a sample and maximum-payoff pure Nash equilibrium.