Affiliation:
1. Hebrew University of Jerusalem, Jerusalem, Israel
2. Weizmann Institute of Science, Rehovot, Israel
3. Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary
Abstract
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.
Funder
European Research Council
Irit Dinur’s ERC-CoG
Hungarian Academy of Sciences
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Mathematics,Marketing,Economics and Econometrics,Statistics and Probability,Computer Science (miscellaneous)
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