David Steurer

Subexponential Algorithms for Unique Games and Related Problems

We give a sub exponential time approximation algorithm for the \textsc{Unique Games} problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size,

Analytical approach to parallel repetition

n^{\epsilon}

Lower Bounds on the Size of Semidefinite Programming Relaxations

. The approximation guarantee depends on~

Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES

\epsilon

Rounding Semidefinite Programming Hierarchies via Global Correlation

, but not on the alphabet size or the number of variables. We also obtain a sub exponential algorithms with improved approximations for \textsc{Small-Set Expansion} and \textsc{Multicut}. For \textsc{Max Cut}, \textsc{Sparsest Cut}, and \textsc{Vertex Cover}, we give sub exponential algorithms with improved approximations on some interesting subclasses of instances. Khots Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for the \textsc{Unique Games}. While our results stop short of refuting the UGC, they do suggest that \textsc{Unique Games} is significantly easier than NP-hard problems such as \textsc{Max 3Sat}, \textsc{Max 3Lin}, \textsc{Label Cover} and more, that are believed not to have a sub exponential algorithm achieving a non-trivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every

Reductions between Expansion Problems

\epsilon>0

Approximate Constraint Satisfaction Requires Large LP Relaxations

and every regular

Message passing algorithms and improved LP decoding

n

Approximation Limits of Linear Programs (Beyond Hierarchies)

-vertex graph~

Rounding sum-of-squares relaxations

G