Direct Product via Round-Preserving Compression

We obtain a strong direct product theorem for two-party bounded round communication complexity. Let sucr(μ, f, C) denote the maximum success probability of an r-round communication protocol that uses at most C bits of communication in computing f(x, y) when (x, y) ∼ μ. Jain et al. [12] have recently showed that if sucr(μ, f, C) ≤ 2 3 and T (C − Ω(r)) · n r , then sucr(μ , f, T ) ≤ exp(−Ω(n/r)). Here we prove that if suc7r(μ, f, C) ≤ 23 and T (C − Ω(r log r)) · n then sucr(μ , f, T ) ≤ exp(−Ω(n)). Up to a log r factor, our result asymptotically matches the upper bound on suc7r(μ , f, T ) given by the trivial solution which applies the per-copy optimal protocol independently to each coordinate. The proof relies on a compression scheme that improves the tradeoff between the number of rounds and the communication complexity over known compression schemes.