(The two proofs that CLIQ is #P-complete that I discuss in this blog are written up by Lance and myself and are here. I think both are well known but I have not been able to find a writeup,so Lance and I did one.)
I have been looking at #P (see my last blog on it it, here, for a refresher on this topic) since I will be teaching this topic in Spring 2021. Recall
#SAT(\phi) = the number of satisfying assignments for phi
#CLIQ((G,k))= the number of cliques of size \ge k of G
#SAT is #P-complete by a cursory look at the proof of the Cook-Levin theorem.
A function is #P-complete if everything else in #P is Turing-Poly red to it. To show for some set A
in NP, #A is #P-complete one usually uses pars reductions.
I wanted a proof that #CLIQ is #P-complete, so I wanted a parsimonious reduction from SAT to CLIQ (Thats a reduction f: SAT--> CLIQ such that the number of satisfying assignments of phi equals the number of cliques of size \ge k of G.)
I was sure there is such a reduction and that it was well known and would be on the web someplace. So I tried to find it.
I tracked some references to a paper by Janos Simon (see here) where he claims that the reduction of SAT to CLIQ by Karp (see here) was pars. I had already considered that and decided that Karps reduction was NOT pars. I looked at both Simon's paper and Karp's paper to make sure I wasn't missing something (e.g., I misunderstood what Simon Says or what Karp ... darn, I can't think of anything as good as `Simon Says'). It seemed to me that Simon was incorrect. If I am wrong let me know.
Someone told me that it was in Valiant's paper (see here). Nope. Valiant's paper shows that counting the number of maximal cliques is #P-complete. Same Deal Here- if I am wrong let me know. One can modify Valiant's argument to get #CLIQ #P-complete, and I do so in the write up. The proof is a string of reductions and starts with PERM is #P-complete. This does prove that# CLIQ is #P-complete, but is rather complicated. Also, the reduction is not pars.
Lance told me an easy pars reduction that is similar to Karp's non-pars reduction, but it really is NOT Karp's reduction. Its in my write up. I think it is well known since I recall hearing it about 10 years ago. From Lance. Hmmm, maybe its just well known to Lance.
But here is my question: I am surprised I didn't find it on the web. If someone can point to a place on the web where it is, please let me know. In any case, its on the web NOW (my write up) so hopefully in the future someone else looking for it can find it.