Bypassing the Unique Games Conjecture for two geometric problems Yi Wu IBM Almaden Research Based on joint work with Venkatesan Guruswami Prasad Raghavendra Rishi Saket CMU Georgia Tech IBM

Unique Games Conjecture

Max 3 SAT Max 2 SAT Max CutMAX 3CSP Max 4 SAT MAX 2AND 0-EXTENSION Multiway Cut MAX 2SATMAX 2LIN MAX 3SAT MultiCut Implications of UGC For a large class of optimization problems, Semidefinite Programming (SDP) gives the best polynomial time approximation.

Status of the UGC

Skepticism of UGC What if UGC is false? The optimality of SDP may not hold. – very few result on the optimality of SDP without UGC. It is not clear whether Unique Games Conjecture is a necessary assumption for all the hardness results.

Overview of our work For two natural geometric problems, we prove that Semidefinite Programming gives the best polynomial time approximation without assuming UGC. – same UG-hardness results known previously.

Problem 1: Subspace approximation

Special case

Our results

Special case

Previous Result:

Our Result

Remarks on our results

Proof overview for subspace approximation

Main Gadget: Dictator Test

Reduction from Smooth Label Cover

Smooth Label Cover

Rest of the proof Composing the Smooth Label Cover with the dictator test.

Future Work