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Integers can be represented with a pair of smaller numbers. To achieve this I use 2 integer sequences.

Integer Sequence 1

1,1,2,1,2,3,1,2,3,4...

And Integer Sequence 2

1,2,1,3,2,1,4,3,2,1...

This gives me a unique pair of smaller integers for every given integer N Like So

1={1,1},2={1,2},3={2,1},4={1,3},5={2,2},6={3,1}..

I've noticed that when reducing 3-Sat to Clique problems you can make 2 simpler 3-sat problems that combine to resolve the main problem. for example..

3 Sat Problem 1. (x1 or !x2 or x3) ^ (x1 or x3 or !x4) ^ (!x1 or !x2 or x4)

is equivalent to

3 Sat Problem 2a and 2b

(x1 or !x1 or x2) ^ (x1 or x2 or !x1) ^ (!x1 or !x1 or x1)

and (x1 or !x2 or x1) ^ (x1 or x1 or !x3) ^ (!x1 or !x2 or x3)

If you reduce problems 2a and 2b into 2 clique problems then overlay the two solutions onto a clique reduction of the first problem they match up.

Not only that but you can keep reducing till your only working with 2 x variables in any given problem subset.

So you could reduce the problem further to 4 subset problems.

Is it possible any of these findings might speed up NP problem solving. to re-phrase the question.

If I break what I am preposing down into 2 parts the time it takes to convert the 3 sat problem with lots of variables into 2 variable problems and what compute space finding any cliques in a 2 variable 3 sat takes at most. I can already estimate quite well how long it's going to take to get the roughly 2*Log[2,n] output 2 variable 3sat problems.

So what I really want to know is how quickly can you computer 2 variable only 3sat problems or where I might find the latest big O or what compute space it's in.

Quoss Wimblik
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    I don't see how this can be answered in its current form. I don't understand your procedure for converting a 3-sat problem into two simpler 3-sat problems, nor how you plan to use the satisfiability of those two simpler 3-cnf formulas to help you decide the satisfiability of the original 3-cnf formula. You say "these findings" but it is not clear what findings you are referring to; I don't see any findings here. An example is not a substitute for a general statement of a theorem or claim. As such, I'm not sure what you are asking. – D.W. Oct 08 '19 at 01:26
  • By the way, I want to warn you in advance that it is not our goal here to make broad advances to science in a single post. – D.W. Oct 08 '19 at 01:27
  • Regardless of the details, it is unclear if a significant advancement is claimed. Reducing a 3sat problem into 2 problems half the size would be fantastic(and probably fantasy) but reducing 3sat into 2 problems with one less variable each is hardly worth writing home about. – Meir Maor Oct 08 '19 at 08:26
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    The way this post should be structured is in several parts. desctibe algorithm a. taking a problem and produce two problems. b. describe algorithm b taking solution for two sub problems and produce solution for original. Argue for correctness. C. describe the size (preferably worst case) of the sub problems in terms of the original problem. d. Discuss aplicability&implications. – Meir Maor Oct 08 '19 at 08:30
  • "Reducing a 3sat problem into 2 problems half the size would be fantastic(and probably fantasy) but reducing 3sat into 2 problems with one less variable each is hardly worth writing home about.". When dealing with a small number of variables the reduction is small when reducing to smaller subsets, However when dealing with bigger numbers of variables each bifurcation leads to 2 smaller numbers that are roughly half the digit size of the parent variable number although the length of the problem has the same number of clauses. – Quoss Wimblik Oct 08 '19 at 21:42
  • There's an entire area centered around constructing exponential time algorithms for NP-hard problems. See for example this monograph. – Yuval Filmus Oct 10 '19 at 12:43
  • You should compare the running time of your algorithm to the state-of-the-art, and then you will have answered your own question. – Yuval Filmus Oct 10 '19 at 12:44

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