Complexity of approximate graph colouring and approximate graph homomorphism

Determine the computational complexity of the promise CSPs corresponding to approximate graph colouring PCSP(K_c, K_d) (with c ≤ d) and approximate graph homomorphism PCSP(A, B) where A and B are undirected graphs; specifically, classify whether these problems are solvable in polynomial time or are NP-hard (or otherwise) across their parameter ranges.

Background

The paper frames many natural approximation problems as promise CSPs (PCSPs). Two central examples are approximate graph colouring, modeled as PCSP(K_c, K_d), and its generalization, approximate graph homomorphism, modeled as PCSP(A, B) for undirected graphs.

While the non-promise variants (graph colouring and graph homomorphism) have long-standing, complete complexity classifications, their promise counterparts remain unresolved. The authors’ results classify the power of local consistency for approximate graph homomorphism (showing linear width in the nontrivial cases), but they do not resolve the overall computational complexity of these problems.