The log-rank conjecture states that the communication complexity of a boolean matrix A is bounded by a polynomial in the log of the rank of A. Equivalently, it says that the chromatic number of a graph is bounded quasi-polynomially in the rank of its adjacency matrix. This old conjecture is well known among computer scientists and mathematicians, but despite extensive work it is still wide open. We survey results relating to the log-rank conjecture, describing the current state of affairs and collecting related questions. Most of the results we discuss are well known, but some points of view are new. One of our hopes is to paint a path to the log-rank conjecture that is made of a series of smaller questions, which might be more feasible to tackle.

对数秩猜想指出，布尔矩阵A的通信复杂性受 A 的秩对数中的多项式的限制。等价地，它表示图的色数在其邻接矩阵的秩上是拟多项式有界的。这个古老的猜想在计算机科学家和数学家中是众所周知的，但尽管进行了大量的工作，它仍然是开放的。我们调查了与对数秩猜想相关的结果，描述了现状并收集了相关问题。我们讨论的大多数结果都是众所周知的，但有些观点是新的。我们的一个希望是描绘一条通向对数秩猜想的道路，该猜想由一系列较小的问题组成，这可能更容易解决。