Approximability of Multiway Partition

1. Approximability ofMultiway Partition Yi Wu IBM Almaden Research Joint work with AlinaEne and Jan Vondrak

2. Definition of Problems

3. Graph Multiway Cut Input: a graph with terminals.

4. Graph Multiway Cut Input: a graph with terminals. Goal: remove minimum number of edges to disconnect the terminals.

5. Graph Multiway Cut Input: a graph with terminals. Equivalent Goal: divide the graph into parts to minimize number of cross edges.

6. Constraint Satisfaction Problem (CSP) with “” constraint 2 1 3 Equivalent Problem: assign to minimize the satisfied inequality.

7. Approximability of Graph Multiway Cut • Upper bound • -approximation by [Calinescu-Karloff-Rabani,1998] • -approximation by [Karger-Klein-Stein-Thorup-Young, 1999] • Lower bound: assuming Unique Games Conjecture, • NP-hard to get better than -approximation. • an earth mover Linear Programming is optimal polynomial time approximation (the ratio is unknown). [Manokaran-Naor-Raghavendra-Schwartz,2008]

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

9. Variant:Hypergraph Multiway Cut (HMC) • Given a hypergraph and -terminals . Remove the minimum number of edges to disconnect • Approximation equivalent to Node Weighted Multiway Cut [Zhao-Nagamochi-Ibaraki 2005]. • Min-CSP with NAE (Not All Equal) constraint on the edges.

10. Generalization:Submodular Multiway Partition • Given a ground set and some submodular function and terminals . Find set • = V Goal: minimize Hypergraph Multiway Cut is a special case[Zhao-Nagamochi-Ibaraki 2005]. A function is submodular if

11. Another interesting SMP: Hypergraph Multiway Partition • Given a hypergraph graphwith -terminals. • partition the graph into parts. • The cost on each edge is the number of different parts it falls in.

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

13. Our Results

14. Our Results (1) 4/3-approximation for 3-way submodular partion. • There is a -approximation for the general submodular multiway partition. • Previous Work: • -approximation by [Chekuri-Ene, 2011] • -approximation for node weighted/hypergraph multiway cut [Garg-Vazirani-Yannakakais,1994] Based on the half integrality of an LP.

15. Overview of the algorithm • Lovasz Relaxation of Submodular Function: • Variables ( dimensional probability simplex) for any • Lovasz Relaxation is the expected value of for the following construction of : choosing random , assign to if • The rounding is not necessarily feasible as can be assigned to multiple • We can efficiently minimize ]

16. The rounding algorithm • Choose a random • For every , set for (i.e, assign to the -th terminal. • Randomly set all the undecided terminals to a partition. To improve from to , the main technicality is analyzing step 3.

17. Our result (2)matching UG-hardness • It is Unique Games-hard to get better than -approximation for hypergraph multiway cut. • Previous work: UG-hardness (from vertex cover). • We prove that assuming UGC, • The integrality gap of a basic LP (generalizing the earth mover LP) is the approximation threshold. • The integrality gap is . • The LP is also optimal for any CSPs contains constraint.

18. The LP for Hypergraph Multiway Cut For hypergraph with terminals : • Variables: for every and , for every and . • Goal: • Constraint: • For every • For every and • For the -th terminal

19. The LP for general Min-CSP For hypergraph and cost function on each • Variables: for every and , for every and . • Goal: • Constraint: • For every • For every and Optimal LP if the constraint contains .

20. Our Results (3): matching oracle hardness • For the submodular multiway partition problem, it requires exponential number of queries on to get better than approximation. • We prove this by constructing symmetric gap of hypergraph multiway cut. Q: is it a coincident that the oracle hardness is the same as the Unique Games hardness?

21. Symmetric gap for Hypergraph Multiway Cut • The graph is symmetric for any permutation from a permutation group . • Vertices are equivalent if there exists some permutation that • A (fractional) solution is symmetric if it is the same on all equivalent vertices. • Symmetric gap: Let I be a instance . Optimum solution (by independent rounding). Optimum Symmetric solution (by independent rounding).

22. Why study symmetric gap? • Symmetric gap of a CSP implies an oracle hardness result for the submodular generalization of that CSP. • Symmetric gap for Max Cut -hardness for non-montone Submodular Function [Feige-Mirrokni-Vondrak, 2007] • Symmetric gap for NAZ (not all zero)  -hardness for Monotone Submodular Function with cardinality constraint [previously known Nemhauser-Wolsey, 1978] • Symmetric gap for Hypergraph Multiway Cut -Submodular Multiway Partition

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 • We have a -approximation for the general submodular multiway partition problem. • oracle hardness and UG-hardness for the hypergraph multiway cut/Node Multiway Cut problem • Equivalence between LP gap and approximation threshold as well as oracle hardness for general CSPs.

25. Open problem • The integrality gap of hypergraph multiway partition? • Between and • It is corresponding to an oracle hardness result for Symmetric submodular multiway partition.