  We design approximation algorithms for Unique Games when the constraint graph
admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem
in $K_r$-minor free graphs, when there is an assignment satisfying
$1-\varepsilon$ fraction of constraints, we present an algorithm that produces
an assignment satisfying $1-O(r\varepsilon)$ fraction of constraints, with the
approximation ratio independent of the alphabet size. A corollary is an
improved approximation algorithm for the ${\sf MaxCut}$ problem for $K_r$-minor
free graphs. For general Unique Games in $K_r$-minor free graphs, we provide
another algorithm that produces an assignment satisfying $1-O(r
\sqrt{\varepsilon})$ fraction of constraints.
  Our approach is to round a linear programming relaxation to find a minimum
subset of edges that intersects all the inconsistent cycles. We show that it is
possible to apply the low diameter graph decomposition technique on the
constraint graph directly, rather than to work on the label extended graph as
in previous algorithms for Unique Games. The same approach applies when the
constraint graph is of genus $g$, and we get similar results with $r$ replaced
by $\log g$ in the ${\sf Max2Lin}_k$ problem and by $\sqrt{\log g}$ in the
general problem. The former result generalizes the result of Gupta-Talwar for
Unique Games in the ${\sf Max2Lin}_k$ case, and the latter result generalizes
the result of Trevisan for general Unique Games.
