A near-optimal direct-sum theorem for communication complexity
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We show a near optimal direct-sum theorem for the two-party randomized communication complexity. Let f⊆ X × Y× Z be a relation, ε> 0 and k be an integer. We show, R^pub_ε(f^k) ·log(R^pub_ε(f^k)) ≥Ω(k ·R^pub_ε(f)) , where f^k= f ×…× f (k-times) and R^pub_ε(·) represents the public-coin randomized communication complexity with worst-case error ε. Given a protocol 𝒫 for f^k with communication cost c · k and worst-case error ε, we exhibit a protocol 𝒬 for f with external-information-cost O(c) and worst-error ε. We then use a message compression protocol due to Barak, Braverman, Chen and Rao [2013] for simulating 𝒬 with communication O(c ·log(c· k)) to arrive at our result. To show this reduction we show some new chain-rules for capacity, the maximum information that can be transmitted by a communication channel. We use the powerful concept of Nash-Equilibrium in game-theory, and its existence in suitably defined games, to arrive at the chain-rules for capacity. These chain-rules are of independent interest.
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