From information to exact communication

We develop a new local characterization of the zero-error information complexity function for two-party communication problems, and use it to compute the exact internal and external information complexity of the 2-bit AND function: IC(AND, 0) = C∧ ≈ 1.4923 bits, and IC^ext(AND, 0) = log₂ 3 ≈ 1.5839 bits. This leads to a tight (upper and lower bound) characterization of the communication complexity of the set intersection problem on subsets of {1, . . . , n} (the player are required to compute the intersection of their sets), whose randomized communication complexity tends to C∧ · n ± o(n) as the error tends to zero. The information-optimal protocol we present has an infinite number of rounds. We show this is necessary by proving that the rate of convergence of the r-round information cost of AND to IC(AND, 0) = C∧ behaves like Θ(1/r ²), i.e. that the r-round information complexity of AND is C∧ +Θ(1/r²). We leverage the tight analysis obtained for the information complexity of AND to calculate and prove the exact communication complexity of the set disjointness function Disj_n(X, Y) = ¬ ∨ⁿ_{i =1} AND(x_i, y_i) with error tending to 0, which turns out to be = CDISJ · n ± o(n), where C_DISJ ≈ 0.4827. Our rate of convergence results imply that an asymptotically optimal protocol for set disjointness will have to use ω(1) rounds of communication, since every rround protocol will be sub-optimal by at least Ω(n/r²) bits of communication. We also obtain the tight bound of 2/ ln 2k ± o(k) on the communication complexity of disjointness of sets of size ≤ k. An asymptotic bound of Θ(k) was previously shown by Håstad and Wigderson.