1. Yi Wu IBM Almaden Research Joint work with Alina Ene and Jan Vondrak

2. Definition of Problems

3. Graph Multiway Cut

4. Goal: remove minimum number of edges to disconnect the terminals.

5. Graph Multiway Cut

6. 1 2 3

7. Approximability of Graph Multiway Cut

8. Variant: Node Weighted Multiway Cut Goal: remove minimum number (weights) of nodes to disconnect the terminals.

9. Variant: Hypergraph Multiway Cut (HMC)

10. Generalization: Submodular Multiway Partition

11. Another interesting SMP: Hypergraph Multiway Partition

12. Relationship Submodular Multiway Partition Hypergraph Multiway cut = Node Weighted Multiway Cut Hypergraph Multiway Partition. Graph Multiway Cut

13. Our Results

14. Our Results (1) 4/3-approximation for 3- way submodular partion. Based on the half integrality of an LP.

15. Overview of the algorithm

16. The rounding algorithm

17. Our result (2) matching UG-hardness

18. The LP for Hypergraph Multiway Cut

19. The LP for general Min-CSP

20. Our Results (3): matching oracle hardness Q: is it a coincident that the oracle hardness is the same as the Unique Games hardness? Q: is it a coincident that the oracle hardness is the same as the Unique Games hardness?

21. Symmetric gap for Hypergraph Multiway Cut Optimum Symmetric solution (by independent rounding). Optimum solution (by independent rounding).

22. Why study symmetric gap?

23. Our Results (4) Q: is it a coincident that the oracle hardness is the same as the Unique Games hardness? A: No. We prove that for any CSP instance, symmetric gap = LP integrality gap.

24. Conclusion

25. Open problem