List of things for Andreas to do

[ndattani]

• I'd like to add into the abstract that the number of auxiliaries for the degree-k monomial is between ceil(log2(k/4)) and floor(log2(k/2)) as opposed to ceil(log2(k/2)). Please try to do this.
• There is a part of the paper where I mention the "Volume 3" book, so that PDF will have to be placed on arXiv soon so that it can be cited properly. Please make sure Volume 3 is updated with every multi-run quad you've found so far, as this will need to have enough substance to pass the tests that the arXiv curators will hold it to
• There is a part of the paper where I say that "one of the quads" presented in this paper took 3.52s to find. Instead of saying it like that, I'd like instead to give the amount of time it took for each multi-run quad presented in our paper. Therefore please give me the runtimes (directly from the program's output) for each of the quads found in this paper, and the output file that shows the quad found and the amount of CPU time it took. We should probably also have in the repository, a copy of the code that was used to find the quads that we present in the paper. This way everything in our paper is easily reproducible by others.
• There's a section with title "Envelope quads with effectively a negative number of auxiliary variables" ... please fill this section in! I have already given the outline.
• The Table in Appendix B should look as beautiful as the table in Appendix A. Usually I do these tedious things myself, but I think it would be good practice for you to try this, which will also help you to appreciate all the work that goes into making a fully presentable paper.

1. There's buttons at the bottom of the LyX window for adding the lines in the table.
2. You can copy and paste the horizontal rule lines that are above and below the table in Appendix A.
3. Some numbers will have to be changed to light grey font to match the table in Appendix A, for this you highlight the text, right-click and change the color to LIGHT grey. Making the numbers bold and blue is also done in the same way.
4. Let's see if you can get the first row of the table to be like the table in Appendix A (add another row using the buttons at the bottom, and then make the first few columns "multi-row", using the buttons at the bottom).
5. See the table in Appendix A and the spacings that I added, to make sure everything looks all nice and centred. You can copy and paste those into the table for Appendix B, but be careful to make sure you put the right amount. For example I have 5 spaces on the left of the left-most column, and only 4 on the right side of the last binary variable. For the f1 and f2 and min(f1,f2) columns, I have one space to the right of most numbers, and two spaces when the number is negative. For the f column, this becomes 2 spaces for most numbers, and 3 spaces when the number is negative.
   Let's see how you get on with this!

• In the section "Application to a computer vision algorithm", you might have noticed that, even back in October (before we started talking about the FoE paper) that I had cited the FoE paper here. Now that you are reading that paper for the computer vision project at Surrey, you have to learn that material anyway, and I'd like you to provide a real example of a computer vision QUBO problem, written out in a form like: 5b1b2b3b4 - 4b3b4b5b6, so that it motivates exactly what we're doing in that section. We'll include this "real" example in the paper, in addition to what we have right now.

[ndattani]

• Nike and Andreas can discuss specific examples of negative aux, so that we don't forget these in the future (Nike was trying to write this on three occasions and never found it easy, so it wouldn't be bad to invest the energy into writing it down properly once and for all), although this might not be so important compared to getting other papers done.
• Andreas to try for the computer vision problem, the best possible FGBZ and pairwise cover.
• Nike: Explain how 8-var/1-aux was created and mention this in the "speed" section
• Andreas: "which does not take any CPU time" I would rather give the exact number in ms.
• Nike: Split-reduc was "last resort" but other envelope quads are "first resort"
• ̶A̶n̶d̶r̶e̶a̶s̶:̶ ̶c̶a̶n̶ ̶y̶o̶u̶ ̶g̶i̶v̶e̶ ̶m̶e̶ ̶t̶h̶e̶ ̶a̶m̶o̶u̶n̶t̶ ̶o̶f̶ ̶t̶i̶m̶e̶ ̶i̶t̶ ̶t̶o̶o̶k̶ ̶f̶o̶r̶ ̶e̶a̶c̶h̶ ̶q̶u̶a̶d̶ ̶p̶r̶e̶s̶e̶n̶t̶e̶d̶ ̶i̶n̶ ̶t̶h̶e̶ ̶p̶a̶p̶e̶r̶?̶ ̶C̶a̶n̶ ̶y̶o̶u̶ ̶a̶l̶s̶o̶ ̶s̶h̶o̶w̶ ̶u̶s̶ ̶a̶n̶ ̶i̶n̶p̶u̶t̶ ̶a̶n̶d̶ ̶o̶u̶t̶p̶u̶t̶ ̶f̶i̶l̶e̶ ̶t̶h̶a̶t̶ ̶r̶e̶p̶r̶o̶d̶u̶c̶e̶s̶ ̶e̶x̶a̶c̶t̶l̶y̶ ̶t̶h̶e̶ ̶q̶u̶a̶d̶s̶ ̶i̶n̶ ̶t̶h̶i̶s̶ ̶p̶a̶p̶e̶r̶,̶ ̶a̶n̶d̶ ̶s̶h̶o̶w̶ ̶t̶h̶e̶ ̶r̶e̶s̶u̶l̶t̶s̶ ̶o̶f̶ ̶t̶h̶e̶ ̶v̶e̶r̶i̶f̶i̶c̶a̶t̶i̶o̶n̶ ̶c̶h̶e̶c̶k̶?̶
• Nike will add Book Volume 3 to arXiv
• Andreas: will check if there's a way to do the computer vision application analytically, otherwise it has to be switched back to the RL because it's wrong to use the "generalized formula" when there's a negative in the trinomial part.
• A̶n̶d̶r̶e̶a̶s̶ Nike will check the 1965 paper to see what the quadratization is.
• Nike will look at the green circled part on pg 3 of his printed version. I think I just have to add a similar paragraph to what I gave for the subsequent section.
• Nike will move footnote to an appendix
• Andreas: look at Table 2 and the whitespace on the right and left of where it says "Eq. 10" in the first row. Compare this to my Table 1.
• Nike: "would results in the following minima" is only true if you get the exact ground state.
• Paper says split-reduc is a special case in intro, but doesn't explain how.
• Nike: "such as the classical algorithm ‘QPBO’ and extensions of it [10], or quantum annealing using thousands of qubits [11] connected by graphs as complicated as Pegasus [12, 13]" we can also mention classical annealers, coherent ising machine, LHZ
• think about changing "cost factor" to "search space"