Brakensiek–Guruswami conjecture on a complexity dichotomy for approximate graph homomorphism

Prove the Brakensiek–Guruswami conjectured dichotomy for approximate graph homomorphism: for undirected graphs A → B, establish that PCSP(A, B) is solvable in polynomial time if A is bipartite or B has a loop, and NP-hard otherwise.

Background

The authors’ main theorem provides a complete classification of the power of local consistency (width) for approximate graph homomorphism, which supports but does not resolve the overall complexity dichotomy conjectured by Brakensiek and Guruswami.

By showing linear width in the nontrivial cases, the paper’s results are consistent with the conjecture’s NP-hard side, while known tractable cases (A bipartite or B looped) remain aligned with the conjecture’s P side.