Brakensiek–Guruswami conjecture on approximate graph homomorphism

Establish the Brakensiek–Guruswami conjectured complexity dichotomy for approximate graph homomorphism PCSP(A, B): prove that the problem is solvable in polynomial time when the source graph A is bipartite or the target graph B has a loop, and NP-hard otherwise.

Background

Approximate graph homomorphism generalizes approximate graph colouring and is a canonical PCSP capturing many approximation tasks on graphs. Despite substantial progress on relaxations and structural algorithms, its full complexity classification remains unsettled.

The conjecture proposed by Brakensiek and Guruswami postulates a clean dichotomy matching structural properties of A and B (bipartiteness of A or presence of a loop in B) with tractability, and NP-hardness otherwise. The authors’ main result on linear width aligns with and supports this conjecture, but does not resolve it.