The Hylland--Zeckhauser (HZ) mechanism is a foundational framework for allocating indivisible goods in one-sided matching markets via lotteries supported by competitive prices and equal artificial budgets. We resolve the exact computational complexity of the HZ scheme by proving that computing an exact HZ equilibrium is FIXP-complete, thereby settling a conjecture of Vazirani and Yannakakis. Notably, this hardness persists even when computing the equilibrium allocation alone. We establish our hardness result via a reduction that encodes quadratic fixed-point equations into a reciprocal variational inequality. On the algorithmic side, we present a deterministic polynomial-time approximation algorithm that achieves an additive error of $0.0755211 + δ$ for any $δ> 0$, significantly improving upon the previous $1/e$ guarantee.
Autobidding equilibria link advertisers' value multipliers or participation probabilities through aggregate budget and return-on-investment (ROI) constraints. We establish the exact computational complexity of finding these equilibria across several canonical settings: exact equilibrium computation is $\mathsf{FIXP}$-complete for second-price ROI pacing, budget- or ROI-constrained generalized second-price (GSP) sponsored shopping, and second-price throttling. Furthermore, approximating the equilibrium control vector in distance to a true equilibrium is $\mathsf{FIXP}_{a}$-complete when the target accuracy is provided as input.