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

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Presentation transcript:

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

Definition of Problems

Graph Multiway Cut

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

Graph Multiway Cut

1 2 3

Approximability of Graph Multiway Cut

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

Variant: Hypergraph Multiway Cut (HMC)

Generalization: Submodular Multiway Partition

Another interesting SMP: Hypergraph Multiway Partition

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

Our Results

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

Overview of the algorithm

The rounding algorithm

Our result (2) matching UG-hardness

The LP for Hypergraph Multiway Cut

The LP for general Min-CSP

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?

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

Why study symmetric gap?

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.

Conclusion

Open problem