Monday, June 12, 2017

Has the Feder-Vardi dichotomy conjecture been proved? (Take 3)

In this post and this one,, I mentioned a paper by Arash Rafiey, Jeff Kinne and Tomás Feder and one by  Andrei Bulatov, claiming a solution to the long-standing Feder-Vardi dichotomy conjecture. Today Moshe Vardi pointed out to me this paper by Dmitriy Zhuk that offers a third proof of that conjecture.

At this point, I think that one can safely say that the dichotomy conjecture is true. Indeed, I find it hard to believe that all the three proofs contain mistakes that cannot be patched. As I wrote in my earlier posts on this matter, I hope all the proofs will be found to be correct by the community and that the techniques used in those articles will find application in other contexts.

Congratulations to the authors of those papers!

1 comment:

Petar Marković said...

As several people in the community have been aware of, the proof of Feder, Kinne and Rafiey has been suspect since it appeared. The parts where they are unclear, or wave hands over details, are precisely the parts where, in the opinion of several experts, the main difficulty was. Unfortunately, this has led to a counterexample and the retraction by Feder, Kinne and Rafiey of the claim they have solved the Dichotomy Conjecture. Their retraction can be found in the comments which replace the abstract of the fourth and most current version of their paper, see

More concretely, it was pointed out in private conversation of several experts that their proof fails on Miklós Maróti's ‘tree-on-Mal’cev’ kind of problems, when they are eliminating what they call `non-minority cases'.

Incidentally, ‘tree-on-Mal’cev’ is not some obscure class of CSPs. As people who are reading it will surely agree, generalizing Maroti's result to 'semilattice-on-Mal’cev’ is the cornerstone of Andrei Bulatov's proof of the Dichotomy Conjecture, the rest is a (very technical and difficult) generalization of the ideas which solve this special case. Even though Feder, Kinne and Rafiey were working only on digraph templates, while ‘tree-on-Mal’cev’ does not specify the relations, just the polymorphisms, the construction by Bulin, Delić, Jackson and Niven, which improves on the original one by Feder and Vardi, reduces a general template to a digraph one, while preserving not only the complexity but also most polymorphisms. So Feder, Kinne and Rafiey's algorithm should have been able to solve those ‘tree-on-Mal’cev’ templates.

Ross Willard, who also was among the experts involved in the initial conversation in January, took the trouble to actually construct digraph CSPs out of the 'tree-on-Mal'cev' CSPs. He constructed out of a tree-on-Mal’cev template a digraph template which contradicts the claim in Feder, Kinne and Rafiey paper that certain situation in the digraphs allows the domain of a variable, and thus the multi-sorted instance, to be reduced by a vertex. This contradicts the whole philosophy of their approach, creating a barrier where they can't proceed with reductions. He emailed the authors of Feder/Kinne/Rafiey his counterexample and, working together, they simultaneously published the retraction, while he published his counterexample on arxiv. The counterexample is available at

On the flip side, at a fairly large conference in June in Novi Sad, Serbia, both Bulatov and Dmitriy Zhuk had plenary talks. After the afternoon in which they exposed their proofs of the Dichotomy Conjecture to a general audience, a special event was organized which lasted almost three hours. The idea was that each would discuss minutiae of their proofs and answer challenges from present experts. In the audience there were about a dozen people who may safely be called experts on the algebraic approach to CSP and most have read in detail large parts of both proofs. There were many interruptions and serious questions were asked, but both proofs survived all challenges unscathed. My degree of confidence after this event in both Bulatov's and Zhuk's proofs is very high, well over 90%, and the odds that both are wrong are really small.

Petar Marković