Paths Beyond Local Search: A Tight Bound for Randomized Fixed-Point Computation

In 1983, Akhus proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid [1 : n]^d from ominus(n^d-1) to 0(d^1/2n^d/2). It remains open whether randomisation helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (Omega(n))^d-1 on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid [1 : n]^d. Our result can be extended to the black-box query model for Sperner´s I&mma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over [1 : n]^d is ominus(n^d), the randomized query model over [ 1 : n]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is ominus(n^d-1).