ECCC-Report TR20-093https://eccc.weizmann.ac.il/report/2020/093Comments and Revisions published for TR20-093en-usThu, 08 Jul 2021 18:28:13 +0300
Revision 1
| Reduction From Non-Unique Games To Boolean Unique Games |
Dana Moshkovitz,
Ronen Eldan
https://eccc.weizmann.ac.il/report/2020/093#revision1We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap $1-\delta$ vs. $1-C\delta$, for any $C> 1$, and sufficiently small $\delta>0$) to the problem of proving a PCP Theorem for a certain non-unique game.
In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness).
The current work is the first to provide an efficient reduction along with a proof of soundness.
The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known.
Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.
Thu, 08 Jul 2021 18:28:13 +0300https://eccc.weizmann.ac.il/report/2020/093#revision1
Paper TR20-093
| Reduction From Non-Unique Games To Boolean Unique Games |
Dana Moshkovitz,
Ronen Eldan
https://eccc.weizmann.ac.il/report/2020/093We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap $1-\delta$ vs. $1-C\delta$, for any $C> 1$, and sufficiently small $\delta>0$) to the problem of proving a PCP Theorem for a certain non-unique game.
In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness).
The current work is the first to provide an efficient reduction along with a proof of soundness.
The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known.
Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.
Tue, 23 Jun 2020 17:41:45 +0300https://eccc.weizmann.ac.il/report/2020/093