Scale-Free Random SAT Instances

Round 1

Reviewer 1 Report

This paper is a theoretical study of SAT formulas modeled as scale-free graphs. The authors review the basic theory of scale-free graphs, percolation and robustness, and apply that framework to discuss the satisfiability of random formulas. The research is original, technically sound, and interesting to understand industrial problems with SAT.  The presentation is good and the illustrations support the theorems and derivations contained in the work.

As minor points, the authors could:

· Omit or cut the discussion on generative models for scale-free graphs (e.g., preferential attachment, configuration model) since eventually a static model is used in the paper. 

· Similarly, shorten the presentation of the Reed-Molloy criterion and its derivation, since it is a well-known result and it is presented in the paper in two complementary forms. The paper is already quite long and some conciseness would be appreciated.

Other than these minor considerations, the paper is recommended for publication.

Author Response

Dear referee,
Thank you for your review. 
We agree with your assessment that the paper is too long, although this is a common "flaw" of most journal articles in computer science, at least compared to other communities (such as physics).

The section on generative models of graphs without scale is not essential. In fact, it is part of the introduction. However, we believe that it is a compact summary of all (most of) the existing models seen from a computer science point of view. We would have liked to find a similar summary when we started. That's why we include it. If you insist on the need to shorten the article, we can move it to an appendix (if the editorial policy allows it) or even delete it.

A similar way applies to the adaptation of Reed-Molloy/Cohen et al. proof. It is a clear example of an independently proven result in two communities, with clearly different styles (rigor/intuition). Originally, we only adapted the Cohen proof, but after feedback from some computer scientists/mathematicians, we decided to also include an adaptation of the Reed-Molloy proof in an attempt to meet the expectations of both types of potential readers.

We have made minor changes to improve the language, following the other referee's suggestions, and shortened the paper.

Reviewer 2 Report

The authors provide a generation model of SAT instances, called scale-free random SAT instances. It is based on the use of a non-uniform probability distribution. The results in formulas where variables' number of occurrences k follows a power-law distribution. The authors prove the existence of an SAT-UNSAT phase transition phenomenon for scale-free random 2-SAT instances. The authors also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the unsatisfiability of most formulas may be due to small cores of clauses. Finally, the authors show how this model will allow us to generate random instances similar to industrial instances of interest for testing purposes. The paper is nicely written. However, some shortcomings must be eliminated before accepting. The list of comments is as follows:

1. The paper is hard to read for people who are not experts;

2. In the intro more human-friendly description of the contribution and problem is needed

3. The formulas should be numerated

4. In all figures, the grid is missing. Moreover, it sometimes cannot be readable enough. 

5. The future research directions should be described in the Conclusion section.

Author Response

Dear referee,
Thank you for your review. 
We have tried to make some changes in the direction you suggested. We number all relevant formulas, leaving unnumbered only intermediate formulas on proofs. 
We agree that some parts are challenging to read, but (after the feedback from some mathematicians) we tried to be rigorous in the proofs (but trying to isolate these more dense parts to allow the reader to skip them).
We do not understand what you mean by the grid. We try to be explicit on the axes of figures, but we think that adding a grid may make figures more unreadable. Anyway, there is no problem in adapting the figures or re-generating them (e.g. avoiding the use of parallel figures) following the editorial guidelines.
The other referee asks us to shorten the paper, especially in two sections. We are also open to moving these parts to an appendix or even removing them, according to the editorial guidelines.