We present new results on approximate colourings of graphs and, more
generally, approximate H-colourings and promise constraint satisfaction
problems.
First, we show NP-hardness of colouring k-colourable graphs with
(⌊k/2⌋k)−1 colours for every k≥4. This improves
the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness
of colouring k-colourable graphs with 2k−1 colours for k≥3, and the
result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring
k-colourable graphs with 2k1/3 colours for sufficiently large k.
Thus, for k≥4, we improve from known linear/sub-exponential gaps to
exponential gaps.
Second, we show that the topology of the box complex of H alone determines
whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite,
H-colourable) G. This formalises the topological intuition behind the result of
Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is
NP-hard for all (3-colourable, non-bipartite) G. We use this technique to
establish NP-hardness of H-colouring of G-colourable graphs for H that include
but go beyond K3, including square-free graphs and circular cliques (leaving
K4 and larger cliques open).
Underlying all of our proofs is a very general observation that adjoint
functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202