If an optimal (even intractable) protocol achieves utility α in k bits, a polynomial-time algorithm can find a protocol achieving α−ε using 2^{O(k)}/ε^2 bits, and this is tight up to a constant in the exponent.
A direct proof for Lovett's bound on the communication complexity of low rank matrices
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abstract
The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O(r^1/2 * log r). Lovett's proof is based on known estimates on the discrepancy of low-rank matrices. We give a simple, direct proof based on a hyperplane rounding argument that in our opinion sheds more light on the reason why a root factor suffices and what is necessary to improve on this factor.
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Computationally Efficient Collaborative Communication Via Regularity-Based Coarsening
If an optimal (even intractable) protocol achieves utility α in k bits, a polynomial-time algorithm can find a protocol achieving α−ε using 2^{O(k)}/ε^2 bits, and this is tight up to a constant in the exponent.