  The UNIQUE GAMES problem is a central problem in algorithms and complexity
theory. Given an instance of UNIQUE GAMES, the STRONG UNIQUE GAMES problem asks
to find the largest subset of vertices, such that the UNIQUE GAMES instance
induced on them is completely satisfiable. In this paper, we give new
algorithmic and hardness results for STRONG UNIQUE GAMES. Given an instance
with label set size $k$ where a set of $(1 - \epsilon)$-fraction of the
vertices induce an instance that is completely satisfiable, our first algorithm
produces a set of $1 - \widetilde{O}({k^2}) \epsilon \sqrt{\log n}$ fraction of
the vertices such that the UNIQUE GAMES induced on them is completely
satisfiable. In the same setting, our second algorithm produces a set of $1 -
\widetilde{O}({k^2}) \sqrt{\epsilon \log d}$ (here $d$ is the largest vertex
degree of the graph) fraction of the vertices such that the UNIQUE GAMES
induced on them is completely satisfiable. The technical core of our results is
a new connection between STRONG UNIQUE GAMES and Small-Set-Vertex-Expansion in
graphs. Complementing this, assuming the Unique Games Conjecture, we prove that
it is NP-hard to compute a set of size larger than $1 - \Omega( \sqrt{\epsilon
\log k \log d})$ for which all the constraints induced on this set are
satisfied.
  Given an undirected graph $G(V,E)$ the ODD CYCLE TRANSVERSAL problem asks to
delete the least fraction of vertices to make the induced graph on the
remaining vertices bipartite. As a corollary to our main algorithmic results,
we obtain an algorithm that outputs a set $S$ such the graph induced on $V
\setminus S$ is bipartite, and $|S|/n \leq O(\sqrt{\epsilon \log d})$ (here $d$
is the largest vertex degree and $\epsilon$ is the optimal fraction of vertices
that need to be deleted). Assuming the Unique Games Conjecture, we prove a
matching (up to constant factors) hardness.
