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### Paper:

TR16-165 | 30th October 2016 15:05

#### Lower Bounds for Elimination via Weak Regularity

TR16-165
Publication: 30th October 2016 15:15
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let $f:\{0,1\}^n \to \{0,1\}$ be any boolean function. Alice and Bob get $k$ inputs $x_1,\ldots,x_k$ and $y_1,\ldots,y_k$ respectively, with $x_i,y_i \in \{0,1\}^n$. They want to output a $k$-bit vector $v$, such that there exists one index $i$ for which $v_i \ne f(x_i,y_i)$. We prove a general result lower bounding the randomized communication complexity of the elimination problem for $f$ using its discrepancy. Consequently, we obtain strong lower bounds for functions Inner-Product and Greater-Than, that work for exponentially larger values of $k$ than the best previous bounds.