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Revision #1 to TR17-148 | 7th October 2017 16:51
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#### The Choice and Agreement Problems of a Random Function

**Abstract:**
The direct-sum question is a classical question that asks whether

performing a task on $m$ independent inputs is $m$ times harder

than performing it on a single input. In order to study this question,

Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose

one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output

a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing

the original task on a single instance, and it is natural to ask whether

it is strictly easier or not. In particular, proving that the choice

problem is not easier is necessary for proving a direct-sum theorem,

and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,

if $f$ is a random function then with high probability its corresponding

choice and agreement problem are not asymptotically easier than computing $f$

on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).

**Changes to previous version:**
Added references to [JKS10] and to [CDKLM17].

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TR17-148 | 6th October 2017 13:23
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#### The Choice and Agreement Problems of a Random Function

**Abstract:**
The direct-sum question is a classical question that asks whether

performing a task on $m$ independent inputs is $m$ times harder

than performing it on a single input. In order to study this question,

Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose

one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output

a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing

the original task on a single instance, and it is natural to ask whether

it is strictly easier or not. In particular, proving that the choice

problem is not easier is necessary for proving a direct-sum theorem,

and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,

if $f$ is a random function then with high probability its corresponding

choice and agreement problem are not asymptotically easier than computing $f$

on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).