1. Lasserre Hierarchy, Higher Eigenvalues and Approximation Schemes for Graph Partitioning and PSD QIP Ali Kemal Sinop (joint work with Venkatesan Guruswami) 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 24:49 PM

3. Minimum Bisection 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. 3 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 Cost=2 µ 4:49 PM

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. OPTAlgorithmα OPT 0 Relaxation 4 1 4:49 PM

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. 54:49 PM

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. 64:49 PM

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 74:49 PM

8. Graph Spectrum and Eigenvalues 1 1 2 2 3 3 4 4 rows and cols indexed by V 8 λ 2 : Measures expansion of the graph through Cheegers inequality. λ r : Related to small set expansion [Arora, Barak, Steurer10], [Gharan, Trevisan11]. 0 = λ 1 λ 2 … λ n 2 and λ 1 + λ 2 + … + λ n = n, 4:49 PM

9. Related Previous Work (Minimization form of) Unique Games (k- labeling with permutation constraints): – [AKKSTV08], [Makarychev, Makarychev10] Constant factor approximation for Unique Games on expanders in polynomial time. – [Kolla10] Constant factor when λ r is large. [Arora, Barak, Steurer10] – For Unique Games and Small Set Expansion, factor in time – For Sparsest Cut, factor assuming 94:49 PM

10. Our Results (1) In time we obtain Why approximation scheme? – 0 = λ 1 λ 2 … λ n 2 and λ 1 + λ 2 + … + λ n = n, 10 Minimum Bisection* Small Set Expansion* Uniform Sparsest Cut Their k-way generalizations* Minimum Bisection* Small Set Expansion* Uniform Sparsest Cut Their k-way generalizations* Independent Set * Satisfies constraints within factor of For r=n, λ r >1, λ n-r <1 Minimum Uncut 4:49 PM

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, Raghavendra11] obtained factor in time using a similar rounding. 114:49 PM

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 124:49 PM

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. 134:49 PM

14. Lasserre Hierarchy Basic idea: Rounding a convex relaxation of minimum bisection. [Lasserre01] 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. 144:49 PM

15. Previous Work on Lasserre Hierarchy Few algorithmic results known before, including: – [Chlamtac07], [Chlamtac, Singh08] n(1) approximation for 3-coloring and independent set on 3-uniform hypergraphs, – [Karlin, Mathieu, Nguyen10] (1+1/r) approximation of knapsack for r-rounds. Known integrality gaps are: – [Schoenebeck08], [Tulsiani09] Most NP-hardness results carry over to (n) rounds of Lasserre. – [Guruswami, S, Zhou11] Factor (1+α) integrality gap for(n) rounds of min-bisection and max-cut. Not ruled out yet: 5-rounds of Lasserre relaxation disproves Unique Games Conjecture. 154:49 PM

16. Why So Few Positive Results? For regular SDP [Goemans, Williamson95] showed that with hyperplane rounding: Prior to our work, no analogue for Lasserre solution. 164:49 PM

17. Consistency Lasserre Relaxation for Minimum Bisection 17 Relaxation for consistent labeling of all subsets of size < r: Marginalization Distribution Partition SIze Cut cost 4:49 PM

18. Rounding Algorithm Choose S with probability – [Deshpande, Rademacher, Vempala, Wang06] 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 [AKKSTTV08] which used propagation from a single node chosen uniformly at random. Return U. 4:49 PM18

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: 19 Normalized Vector for x S (f) OPT 19 4:49 PM

20. Matrix Π S Remember {x S (f)} f are orthogonal. is a projection matrix onto span{x S (f)} f. For any 20 Let P S be the corresponding projection matrix. 4:49 PM

21. Low Rank Matrix Reconstruction The final bound is: For any S of size r this is lower bounded by: [Guruswami, S11] Volume sampling columns yield – And this bound is tight. best rank-r approximation of X 214:49 PM

22. Relating Reconstruction Error to Graph Spectrum Best rank-r approximation is obtained by top r-eigenvectors. Using Courant-Fischer theorem, Therefore 224:49 PM

23. Summary Gave a randomized rounding algorithm based on propagation from a seed set S so that: Related choosing S to low rank matrix reconstruction error. Bounded low rank matrix reconstruction error in terms of λ r. 234:49 PM

24. Questions? Thanks. 244:49 PM