ECCC-Report TR17-030https://eccc.weizmann.ac.il/report/2017/030Comments and Revisions published for TR17-030en-usSat, 18 Feb 2017 17:23:23 +0200
Paper TR17-030
| An Improved Dictatorship Test with Perfect Completeness |
Amey Bhangale,
Subhash Khot,
Devanathan Thiruvenkatachari
https://eccc.weizmann.ac.il/report/2017/030A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.
The dictatorship test is said to have {\em perfect completeness} if it accepts any dictator function. The {\em soundness} of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a $k$-query dictatorship test with perfect completeness and soundness $ \frac{2k + 1}{2^k}$, where $k$ is of the form $2^t -1$ for any integer $t > 2$. This improves upon the result of \cite{TY15} which gave a dictatorship test with soundness $ \frac{2k + 3}{2^k}$.Sat, 18 Feb 2017 17:23:23 +0200https://eccc.weizmann.ac.il/report/2017/030