Al i Kemal Sinop (joint work with Venkatesan Guruswami ) Carnegie Mellon University

1. Lasserre Hierarchy,Higher Eigenvalues and Approximation Schemes for Graph Partitioning and PSD QIP AliKemalSinop (joint work with VenkatesanGuruswami) Carnegie Mellon University

2. Outline • Introduction • Sample Problem: Minimum Bisection • Approximation Algorithms • Our Motivation and Results Overview • Results • Graph Spectrum • Related Work and Our Results • Case Study: Minimum Bisection • Lasserre Hierarchy Formulation • Rounding Algorithm • Analysis

3. Minimum Bisection µ • Given graph G=(V,E,W), find subset of size n/2 which cuts as few edges as possible. • Canonical problem for graph partitioning by allowing arbitrary size: • Small Set Expansion (weight each node by its degree) • Uniform Sparsest Cut (try out all partition sizes in small increments) • Etc… • NP-hard. 1 1 2 2 3 3 4 4 Cost=2

4. Approximation Algorithms • Find an α-factor approximation. • If minimum cost = OPT, • Algorithm always finds a solution with value ≤ α OPT. • (This work) Round a convex relaxation. 0 Relaxation OPT Algorithm α OPT 1

5. Motivation • For many graph partitioning problems (including minimum bisection), huge gap between hardness and approximation results. • Best known algorithms have factor • Whereas no 1.1 factor hardness is known. • We want to close the gap.

6. Our Results: Overview • For graph partitioning problems including: • Minimum bisection, • Small set expansion, • Uniform sparsest cut, • Minimum uncut, • Their k-way generalizations, etc… • We give approximation schemes whose running time is dependent on graph spectrum.

7. Outline • Introduction • Sample Problem: Minimum Bisection • Approximation Algorithms • Our Motivation and Results Overview • Results • Graph Spectrum • Related Work and Our Results • Case Study: Minimum Bisection • Lasserre Hierarchy Formulation • Rounding Algorithm • Analysis

8. Graph Spectrum and Eigenvalues 1 2 3 4 rows and cols indexed by V 0 = λ1 ≤λ2 ≤… ≤ λn ≤2 and λ1+ λ2+ … + λn = n, λ2: Measures expansion of the graph through Cheeger’s inequality . λr: Related to small set expansion [Arora, Barak, Steurer’10], [Gharan, Trevisan’11].

9. Related Previous Work • (Minimization form of) Unique Games (k-labeling with permutation constraints): • [AKKSTV’08], [Makarychev, Makarychev’10] Constant factor approximation for Unique Games on expanders in polynomial time. • [Kolla’10] Constant factor when λr is large. • [Arora, Barak, Steurer’10] • For Unique Games and Small Set Expansion, factor in time • For Sparsest Cut, factor assuming

10. Our Results (1) • In time we obtain • Why approximation scheme? • 0 = λ1 ≤λ2 ≤… ≤ λn ≤2 and λ1+ λ2+ … + λn = n, Minimum Bisection* Small Set Expansion* Uniform Sparsest Cut Their k-way generalizations* Minimum Uncut For r=n, λr >1, λn-r <1 Independent Set * Satisfies constraints within factor of

11. Our Results for Unique Games • For Unique Games, a direct bound will involve spectrum of lifted graph, whereas we want to bound using spectrum of original graph. • We give a simple embedding and work directly on the original graph. • We obtain factor in time . • Concurrent to our work, [Barak, Steurer, Raghavendra’11] obtained factor in time using a similar rounding.

12. Outline • Introduction • Sample Problem: Minimum Bisection • Approximation Algorithms • Our Motivation and Results Overview • Results • Graph Spectrum • Related Work and Our Results • Case Study: Minimum Bisection • Lasserre Hierarchy • Rounding Algorithm • Analysis

13. Case Study: Minimum Bisection • We will present an approximation algorithm for minimum bisection problem on d-regular unweighted graphs. • We will show that it achieves factor . • Obtaining factor requires some additional ideas.

14. Lasserre Hierarchy • Basic idea: Rounding a convex relaxation of minimum bisection. • [Lasserre’01] Strongest known SDP-relaxation. • (Relaxation of) For each subset S of size ≤ r and each possible labeling of S, • An indicator vector which is 1 if S is labeled with f • 0 else. • And all implied consistency constraints.

15. Previous Work on Lasserre Hierarchy • Few algorithmic results known before, including: • [Chlamtac’07], [Chlamtac, Singh’08] nΩ(1) approximation for 3-coloring and independent set on 3-uniform hypergraphs, • [Karlin, Mathieu, Nguyen’10] (1+1/r) approximation of knapsack for r-rounds. • Known integrality gaps are: • [Schoenebeck’08], [Tulsiani’09] Most NP-hardness results carry over to Ω(n) rounds of Lasserre. • [Guruswami, S, Zhou’11]Factor (1+α) integrality gap for Ω(n) rounds of min-bisection and max-cut. • Not ruled out yet: “5-rounds of Lasserre relaxation disproves Unique Games Conjecture.”

16. Why So Few Positive Results? • For regular SDP [Goemans, Williamson’95] showed that with hyperplane rounding: • Prior to our work, no analogue for Lasserre solution.

17. Lasserre Relaxation for Minimum Bisection • Relaxation for consistent labeling of all subsets of size < r: Cut cost Consistency Marginalization Distribution Partition SIze

18. Rounding Algorithm • Choose S with probability • [Deshpande, Rademacher, Vempala, Wang’06]Volume sampling. • Label S by choosing f with probability . • Propagate to other nodes: • For each node v, • With probability include v in U. • Inspired by [AKKSTTV’08] which used propagation from a single node chosen uniformly at random. • Return U.

19. Analysis • Partition Size • Each node is chosen into U independently • By Chernoff, with high probability • Number of Edges Cut • After arithmetization, we have the following bound: Normalized Vector for xS(f) 19 ≤ OPT

20. Matrix ΠS • Remember {xS(f)}f are orthogonal. is a projection matrix onto span{xS(f)}f . • For any Let PS be the corresponding projection matrix.

21. Low Rank Matrix Reconstruction • The final bound is: • For any S of size r this is lower bounded by: • [Guruswami, S’11] Volume sampling columns yield • And this bound is tight. best rank-r approximation of X

22. Relating Reconstruction Error to Graph Spectrum • Best rank-r approximation is obtained by top r-eigenvectors. • Using Courant-Fischer theorem, • Therefore

23. Summary • Gave a randomizedroundingalgorithmbased on propagation from a seed set S so that: • Related choosing S to low rank matrix reconstruction error. • Bounded low rank matrix reconstructionerror in terms of λr.

24. Questions? • Thanks.