Complexity of approximate graph colouring and approximate graph homomorphism

Determine the computational complexity of the approximate graph colouring problem PCSP(K_c, K_d) with c ≤ d and of the approximate graph homomorphism problem PCSP(A, B) for undirected graphs A → B; explicitly classify whether these promise CSPs are solvable in polynomial time or are NP-hard (or otherwise), since their complexity status remains unresolved.

Background

The paper explains that many fundamental problems that are not expressible as standard CSPs can be formulated as promise CSPs (PCSPs). Two primary examples are approximate graph colouring (PCSP(K_c, K_d)) and approximate graph homomorphism (PCSP(A, B) for undirected graphs).

Despite significant progress on CSPs, including dichotomy theorems, the promise counterparts remain largely unsettled from a complexity-theoretic perspective. The authors highlight that, in contrast to non-promise variants whose complexity has long been classified, the exact complexity of these approximate problems is still unresolved.