  A graph homomorphism is a vertex map which carries edges from a source graph
to edges in a target graph. We study the approximability properties of the
Weighted Maximum H-Colourable Subgraph problem (MAX H-COL). The instances of
this problem are edge-weighted graphs G and the objective is to find a subgraph
of G that has maximal total edge weight, under the condition that the subgraph
has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX
k-CUT. To this end, we introduce a metric structure on the space of graphs
which allows us to extend previously known approximability results to larger
classes of graphs. Specifically, the approximation algorithms for MAX CUT by
Goemans and Williamson and MAX k-CUT by Frieze and Jerrum can be used to yield
non-trivial approximation results for MAX H-COL. For a variety of graphs, we
show near-optimality results under the Unique Games Conjecture. We also use our
method for comparing the performance of Frieze & Jerrum's algorithm with
Hastad's approximation algorithm for general MAX 2-CSP. This comparison is, in
most cases, favourable to Frieze & Jerrum.
