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 AN D function: IC(AND, 0) = C ∧ ≈ 1.4923 bits, and IC ext (AND, 0) = log 2 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 AN D to IC(AND, 0) = C ∧ behaves like Θ(1/r 2 ), i.e. that the r-round information complexity of AN D is C ∧ + Θ(1/r 2 ).We leverage the tight analysis obtained for the information complexity of AN D to calculate and prove the exact communication complexity of the set disjointness function Disj n(X, Y ) = ¬ ∨ n i=1 AN D(xi, yi) with error tending to 0, which turns out to be = C DISJ · 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/r 2 ) bits of communication. * In this extended abstract, we omit most proofs, and shorten others. The full version of this paper can be found at ECCC † Research supported in part by an Alfred P. Sloan Fellowship, an NSF CAREER award (CCF-1149888), and a Turing Centenary Fellowship.We also obtain the tight bound of 2 ln 2 k ± 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.
The priority model of "greedy-like" algorithms was introduced by Borodin, Nielsen, and Rackoff in 2002. We augment this model by allowing priority algorithms to have access to advice, i.e., side information precomputed by an all-powerful oracle. Obtaining lower bounds in the priority model without advice can be challenging and may involve intricate adversary arguments. Since the priority model with advice is even more powerful, obtaining lower bounds presents additional difficulties. We sidestep these difficulties by developing a general framework of reductions which makes lower bound proofs relatively straightforward and routine. We start by introducing the Pair Matching problem, for which we are able to prove strong lower bounds in the priority model with advice. We develop a template for constructing a reduction from Pair Matching to other problems in the priority model with advice -this part is technically challenging since the reduction needs to define a valid priority function for Pair Matching while respecting the priority function for the other problem. Finally, we apply the template to obtain lower bounds for a number of standard discrete optimization problems.
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