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# Title: The Proof of CSP Dichotomy Conjecture

Authors: Dmitriy Zhuk
Abstract: Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a core of a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete.
In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.
 Comments: 1) The notion "link" is renamed to "bridge" 2) I never use word "subdirect" for instances. 3) To simplify, I replaced statement of Lemma 7.13 by the statement of Corollary 7.13.1 and removed Corollary 7.13.1. 4) I changed the definition of a weaker constraint. Now it is strictly weaker. 5) I fixed few misprints and added few clarifications (for example in the proof of Lemma 7.10) Subjects: Computational Complexity (cs.CC) Cite as: arXiv:1704.01914 [cs.CC] (or arXiv:1704.01914v5 [cs.CC] for this version)

## Submission history

From: Dmitriy Zhuk [view email]
[v1] Thu, 6 Apr 2017 16:27:29 GMT (17kb,D)
[v2] Sun, 14 May 2017 21:06:40 GMT (56kb,D)
[v3] Thu, 1 Jun 2017 17:28:13 GMT (55kb,D)
[v4] Tue, 13 Jun 2017 20:31:08 GMT (55kb,D)
[v5] Sun, 25 Jun 2017 11:15:29 GMT (57kb,D)
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