The Unique Games Conjecture and Graph Expansion School on Approximability, Bangalore, January 2011 Joint work with S Prasad Raghavendra Georgia Institute of Technology David Steurer Microsoft Research New England

The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

U NIQUE G AMES Goal:satisfy as many constraints as possible

U NIQUE G AMES Goal:satisfy as many constraints as possible Goal: Distinguish two cases Unique Games Conjecture (UGC) [Khot’02]

Implications of UGC For many basic optimization problems, it is NP-hard to beat current algorithms (based on simple LP or SDP relaxations) Examples: V ERTEX C OVER [Khot-Regev’03], M AX C UT [Khot-Kindler-Mossel-O’Donnell’04, Mossel-O’Donnell-Oleszkiewicz’05], every M AX C SP [Raghavendra’08], …

What are hard instances for U NIQUE G AMES ?

Random instances Random 3-S AT Hypothesis [Appelbaum-Barak-Wigderson’10, Bhaskara- -Charikar-Chlamtac-Feige-Vijayaraghavan’10] Other problems: Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :  expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]  few large eigenvalues [Kolla’10, Barak-Raghavendra-S.’11]  strong small-set expanders [Arora-Impagliazzo-Matthews-S.’10] [Feige’02, Schoenebeck’08] Easy for random-like instances

What are hard instances for U NIQUE G AMES ? Random 3-S AT,Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :Easy for random-like instances Other problems: Random instances Other problems: C LIQUE on product graphs P ROJECTION GAMES from parallel repetition [Raz’98] U NIQUE G AMES : Easy for parallel-repeated instances of M AX C UT [Barak-Hardt-Haviv-Rao-Regev-S.’08] (based on counterexample for strong parallel repetition [Raz’08]) Combinations of mildly hard instances

C LIQUE, U NIQUE G AMES : P ROJECTION GAMES Easy for parallel-repeated instances of M AX C UT What are hard instances for U NIQUE G AMES ? Combinations of mildly hard instances Random 3-S AT,Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :Easy for random-like instances Other problems: Random instances Other problems: Hard instances for U NIQUE G AMES from hard instances for S MALL S ET E XPANSION [here] natural generalization of S PARSEST C UT

The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

d-regular graph G d vertex set S Graph Expansion expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| Important notion in many contexts: derandomization, network routing, coding theory, Markov chains, differential geometry, group theory

S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V|

S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G eigenvalue gap: first non-trivial approximation [Cheeger’70] No strong hardness for approximating S PARSEST C UT known! (even assuming UGC) Approximating S PARSEST C UT

S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G S MALL -S ET E XPANSION no poly-time algorithm with non-trivial approximation guarantee known Goal: Distinguish two cases Small Set Expansion Hypothesis (SSE)

S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G S MALL -S ET E XPANSION no poly-time algorithm with non-trivial approximation guarantee known Goal: Distinguish two cases Small Set Expansion Hypothesis (SSE)

The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

“Superficial” Connection of U NIQUE G AMES and S MALL -S ET E XPANSION

SSE UGC + SSE [Raghavendra-S.’10] UGC additional promise: small-set expansion of constraint graph M AX C UT V ERTEX C OVER Any M AX C SP … S PARSEST C UT B ALANCED S EPARATOR M INIMUM L INEAR A RRANGEMENT M IN k-C UT … [Raghavendra- -Tulsiani-S.’10] [Raghavendra- -Tulsiani-S.’10]

graph G A B Verifier BA Prover 1Prover 2 ba Verifier To show: a b

graph G A B

S A B Completeness Partial Strategy for Prover 1 (and 2) : (otherwise, refuse to answer)

graph G Soundness S

The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

Goal: rule out that certain (classes of) algorithms refute the UGC or the SSE hypothesis Here:algorithms based on (hierarchies) of SDP relaxations (capture current best approximation algorithms)

Basic SDP relaxation for S MALL -S ET E XPANSION minimize subject to

Basic SDP relaxation for S MALL -S ET E XPANSION minimize subject to Integrality Gap Instance

Integrality Gaps for U NIQUE G AMES via reductions  super-polynomial lower-bound for U NIQUE G AMES in a restricted computational model

Integrality Gap Instance Integrality Gaps for U NIQUE G AMES via reductions Thank you!Questions?