Log-rank conjecture

In theoretical computer science, the log-rank conjecture states that the deterministic communication complexity of a two-party Boolean function is polynomially related to the logarithm of the rank of its input matrix.

Let $$D(f)$$ denote the deterministic communication complexity of a function, and let $$\operatorname{rank}(f)$$ denote the rank of its input matrix $$M_f$$ (over the reals). Since every protocol using up to $$c$$ bits partitions $$M_f$$ into at most $$2^c$$ monochromatic rectangles, and each of these has rank at most 1,
 * $$D(f) \geq \log_2 \operatorname{rank}(f). $$

The log-rank conjecture states that $$D(f)$$ is also upper-bounded by a polynomial in the log-rank: for some constant $$C$$,
 * $$D(f) = O((\log \operatorname{rank}(f))^C). $$

Lovett proved the upper bound
 * $$D(f) = O\left(\sqrt{\operatorname{rank}(f)} \log \operatorname{rank}(f)\right). $$

This was improved by Sudakov and Tomon, who removed the logarithmic factor, showing that
 * $$D(f) = O\left(\sqrt{\operatorname{rank}(f)}\right). $$

This is the best currently known upper bound.

The best known lower bound, due to Göös, Pitassi and Watson, states that $$C \geq 2$$. In other words, there exists a sequence of functions $$f_n$$, whose log-rank goes to infinity, such that
 * $$ D(f_n) = \tilde\Omega((\log \operatorname{rank}(f_n))^2). $$

In 2019, an approximate version of the conjecture has been disproved.