ECCC-Report TR19-053https://eccc.weizmann.ac.il/report/2019/053Comments and Revisions published for TR19-053en-usTue, 09 Apr 2019 20:18:16 +0300
Paper TR19-053
| The complexity of 3-colouring $H$-colourable graphs |
Andrei Krokhin,
Jakub Opršal
https://eccc.weizmann.ac.il/report/2019/053We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Nešet?il, this problem is NP-hard for any non-bipartite graph $H$. In the context of promise constraint satisfaction problems, Brakensiek and Guruswami conjectured that this hardness result extends to promise graph homomorphism as follows: fix any non-bipartite graph $H$ and another graph $G$ with a homomorphism from $H$ to $G$, it is NP-hard to find a homomorphism to $G$ from a given $H$-colourable graph. Arguably, the two most important special cases of this conjecture are when $H$ is fixed to be $K_3$ (and $G$ is any graph with a triangle) and when $G=K_3$ (and $H$ is any 3-colourable graph). The former case is equivalent to the notoriously difficult approximate graph colouring problem. In this paper, we confirm the Brakensiek-Guruswami conjecture for the latter case. Our proofs rely on a novel combination of the universal-algebraic approach to promise constraint satisfaction, that was recently developed by Bulín and the authors, with some ideas from algebraic topology. Tue, 09 Apr 2019 20:18:16 +0300https://eccc.weizmann.ac.il/report/2019/053