1. 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

2. Unique Games Conjecture

3. 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.

4. Status of the UGC

5. 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.

6. 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.

7. Problem 1: Subspace approximation

8. Special case

9. Our results

11. Special case

12. Previous Result:

13. Our Result

14. Remarks on our results

15. Proof overview for subspace approximation

16. Main Gadget: Dictator Test

19. Reduction from Smooth Label Cover

20. Smooth Label Cover

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

22. Future Work