On Communication Complexity of Fixed Point Computation

AbstractBrouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from \(\) to \(\), and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of \(\) for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model.Each player is given a function from \(\) to \(\), and their goal is to find an approximate fixed point of the concatenation of the functions.Each player is given a function from \(\) to \(\), and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of \(\) for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation \(\) of the \(\)-simplex is publicly known and one player is given a set \(\) and a coloring function from \(\) to \(\), and the other player is given a set \(\) and a coloring function from \(\) to \(\), such that \(\), and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of \(\) for the aforementioned problem as well (when \(\) is large). On the positive side, we show that if \(\) then there is a deterministic protocol for the Sperner problem with \(\) bits of communication.