Brouwer’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
[0,1]^n
to
[0,1]^n
, 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
2^{\Omega (n)}
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
[0,1]^n
to
[0,1]^{n/2}
, and their goal is to find an approximate fixed point of the
concatenation
of the functions.
Each player is given a function from
[0,1]^n
to
[0,1]^{n}
, 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
2^{\Omega (n)}
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
T
of the
d
-simplex is publicly known and one player is given a set
S_A\subset T
and a coloring function from
S_A
to
\lbrace 0,\ldots ,d/2\rbrace
, and the other player is given a set
S_B\subset T
and a coloring function from
S_B
to
\lbrace d/2+1,\ldots ,d\rbrace
, such that
S_A\dot{\cup }S_B=T
, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of
|T|^{\Omega (1)}
for the aforementioned problem as well (when
d
is large). On the positive side, we show that if
d\le 4
then there is a deterministic protocol for the Sperner problem with
O((\log |T|)^2)
bits of communication.