Revised: July 31, 2020

Published: November 9, 2020

**Keywords:**distributions, sampling, correlated sampling, coupling, MinHash, communication complexity

**Categories:**algorithms, approximation algorithms, sampling, distributions, correlated sampling, coupling, communication complexity

**ACM Classification:**F.0, G.3

**AMS Classification:**68Q99, 94A20, 68W15

**Abstract:**
[Plain Text Version]

In the *correlated sampling* problem, two players
are given probability distributions $P$ and $Q$, respectively,
over the same finite
set,
with access to shared randomness.
Without any communication, the two players are each required to output an
element sampled according to their respective distributions, while trying to
minimize the probability that their outputs disagree.
A well known strategy due to Kleinberg--Tardos and Holenstein, with a close variant (for a similar problem) due to Broder, solves this task with disagreement probability at most $2 \delta/(1+\delta)$, where $\delta$ is the total variation distance between $P$ and $Q$. This strategy has been used in several different contexts,
including sketching algorithms, approximation algorithms based on rounding linear programming relaxations, the study of parallel repetition and cryptography.

In this paper, we give a surprisingly simple proof that this strategy is essentially optimal. Specifically, for every $\delta \in (0,1)$, we show that any correlated sampling strategy incurs a disagreement probability of essentially $2\delta/(1+\delta)$ on some inputs $P$ and $Q$ with total variation distance at most $\delta$. This partially answers a recent question of Rivest.

Our proof is based on studying a new problem that we call *constrained agreement*. Here, the two players are given subsets $A \subseteq [n]$ and $B \subseteq [n]$, respectively,
and their goal is to output an element $i \in A$ and $j \in B$, respectively,
while minimizing the probability that $i \neq j$. We prove tight bounds for this question, which in turn imply tight bounds for correlated sampling. Though we settle basic questions about the two problems, our formulation leads to more fine grained questions that remain open.