A Note on the LogRank Conjecture in Communication Complexity

Round 1

Reviewer 1 Report

In this paper, the authors discussed the  LogRank Conjecture in communication complexity. And a communication protocol is presented and analyzed. The work presented in this paper is not sufficient, and the whole manuscript looks top-heavy. Too much discussions in the introductions. 

1. The introduction section could be improved. There are some theorems and preliminaries. I think it is better to put in a seperate section. 

2.  Can the author draw a block diaram based on the protocol in Section 2?

3. Section 2 and Section 3 are not discussed  adequately.

4.  "4. Appendix" can be put in Section 3. And some more discussions should be added. 

Author Response

The author is indebted for the remarks of the Reviewer, Please find enclosed the answers of the helpful suggestions below:

1, We accepted the kind suggestion of the Reviewer, and the Introduction section is now split into "Introduction" and "Preliminaries"

2, Instead of a block diagram, we clearly structured the protocol in the revision.

3, We have given an explicit example now for the coefficients in Theorem 5 in the new section No. 4. Additionally, we have also made available a Maple worksheet for the interested readers with the large matrices.

4, We have moved the Appendix in the main text, and extended it considerably.

Reviewer 2 Report

This paper addresses the log rank conjecture in communication complexity.  

The main result shows that, given a communication function f, there is another

function g somewhat related to f whose communication complexity is upper bounded

by a polynomial in the logarithm of the rank of f.  This result follows 

quite directly from a previous result of the author giving a particular kind 

of polynomial representation of the dot-product function.

Specific comments:

  o Spelling of Lovasz is messed up in abstract and before conjecture 2.

  o Conjecture 2: Some constant needs to be added to the right hand side. When

M_f has rank one, the communication complexity can be non-zero but the right 

hand side is zero.

  o The main result should be stated as a theorem.

Author Response

The author is grateful for the remarks and suggestions of the Reviewer.

We have corrected the spelling of Lovasz.

The rank=0 or 1 cases are taken care for now, thank you for the suggestions.

The main result is now split into a Theorem and a Corollary, plus a 3-point formal protocol.

Round 2

Reviewer 1 Report

The author has revised the mansucript and I think it can be accepted for publciation. 

Author Response

The author is truly grateful for the review.