Effective-Resistance-Reducing Flows, Spectrally
Description: Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and ATSP Nima Anari UC Berkeley Shayan Oveis Gharan Univ of Washington History of Graph Sparsification 2 Spectral Sparsifier Spielman-Teng04 Cut Sparsifier Benczur-Karger96
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slide1. Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and ATSP Nima Anari
UC Berkeley Shayan Oveis Gharan
Univ of Washington<br>
slide2. History of Graph Sparsification 2 Spectral Sparsifier [Spielman-Teng’04] Cut Sparsifier
[Benczur-Karger’96] Spectrally Thin Tree Proof of Kadison-Singer
[Marcus-Spielman-Srivastava’13] Asymmetric TSP
[O-Anari’14] Laplacian Solvers
[Spielman-Teng’04] SS by Effective Resistance [Spielman-Srivastava’09] Linear size SS [Batson-Spielman-Srivastava’10] This Talk<br>
slide3. Asymmetric TSP (ATSP) 3<br>
slide4. Linear Programming Relaxation [Held-Karp’72] 4 Integrality Gap:<br>
slide5. Previous Works Approximation Algorithms
log(n) [Frieze-Galbiati-Maffioli’82]
0.999 log(n) [Bläser’02]
0.842 log(n) [Kaplan-Lewenstein-Shafrir-Sviridenko’05]
0.666 log(n) [Feige-Singh’07]
O(logn/loglogn) [Asadpour-Goemans-Madry-O-Saberi’09]
O(1) (planar/bd genus) [O-Saberi’10,Erickson-Sidiropoulos’13]
Integrality Gap
≥ 2 [Charikar-Goemans-Karloff’06]
≤ O(logn/loglogn) [AGMOS’09]. 5<br>
slide6. Main Result 6 For any cost function, the integrality gap of the LP relaxation is polyloglog(n).<br>
slide7. Plan of the Talk 7 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys<br>
slide8. Thin Spanning Trees 8 Kn 2/n-thin tree Application: f(n)/k-thin trees imply 5f(n)-approx for ATSP<br>
slide9. Main Result 9 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide10. 10 In Pursuit of Thin Trees<br>
slide11. Main Ingredients 11 Interlacing Polynomials and
thin basis problem Effective resistance reduction
by Convex Programming<br>
slide12. A General Framework for L1 Opt Problems 12<br>
slide13. We can use convex duality to analyze Q(.) of the optimum. 13 Has exp. many constraints but can be solved using ellipsoid Since Q(.) is convex the program is convex<br>
slide14. 14<br>
slide15. 15<br>
slide16. Plan of the Talk 16 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys Min Cost
Flow Eff Resist
Reduction<br>
slide17. Thin Basis Prob [Marcus-Spielman-Srivastava’13] 17 d Linearly independent set of vectors<br>
slide18. A Weaker Suff Cond for Thin Basis Problem 18 ………………………………………<br>
slide19. Proof of Thin Basis Thm 19<br>
slide20. Method of Interlacing Polynomials 20<br>
slide21. Summary/Future Directions Main Ingredients:
Effective Resistance Reduction
Thin Basis Problem
Future Directions:
Algorithmic proof of [MSS’13].
Existence of C/k thin trees
Subsequent work: Svensson’s 27-app algorithm for graph ATSP. 21<br>
slide22. Previous Works: Randomized Rounding 22<br>
slide23. Main Result 23 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide24. Previous Works: Randomized Rounding 24<br>
slide25. Graph Laplacian 25 E.g.,<br>
slide26. A Necessary Condition for Spectral Thinness 26 where<br>
slide27. A k-con Graph with no Spectrally Thin Tree 27 n/k vertices k edges k edges<br>
slide28. A Sufficient Condition for Spectral Thinness 28<br>
slide29. Spectrally Thin Trees (Summary) 29 k-edge
connectivity O(1/k)-combinatorial thin tree O(1/k)-spectrally thin tree [MSS13] ?<br>
slide30. Our Approach 30<br>
slide31. Main Idea 31 Symmetrize L2 structure of G
while preserving its L1 structure<br>
slide32. An Example 32 n/k vertices<br>
slide33. An Observation 33<br>
slide34. Main Idea 34 D+G has a spectrally thin tree and
any spectrally thin tree of G+D is (comb) thin in G. Bypasses Spectral Thinness Barrier.<br>
slide35. An Impossibility Theorem 35<br>
slide36. Proof Overview 36 D is not a graph<br>
slide37. Proof Overview 37<br>
slide38. A Weaker Goal: Satisfying Degree Cuts 38<br>
slide39. A Convex Program for Optimum D 39<br>
slide40. Main Result 40 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide41. Conclusion Main Idea:
Symmetrize L2 structure of G while preserving its L1 structure
Tools:
Interlacing polynomials/Real Stable polynomials
Convex optimization
Graph partitioning
High dimensional geometry 41<br>
slide42. Future Works/Open Problems Algorithmic proof of [MSS’13] and our extension.
Existence of C/k thin trees and constant factor approximation algorithms for ATSP.
Subsequent work: Svensson designed a 27-app algorithm for ATSP when c(.,.) is a graph metric. 42<br>
UC Berkeley Shayan Oveis Gharan
Univ of Washington<br>
slide2. History of Graph Sparsification 2 Spectral Sparsifier [Spielman-Teng’04] Cut Sparsifier
[Benczur-Karger’96] Spectrally Thin Tree Proof of Kadison-Singer
[Marcus-Spielman-Srivastava’13] Asymmetric TSP
[O-Anari’14] Laplacian Solvers
[Spielman-Teng’04] SS by Effective Resistance [Spielman-Srivastava’09] Linear size SS [Batson-Spielman-Srivastava’10] This Talk<br>
slide3. Asymmetric TSP (ATSP) 3<br>
slide4. Linear Programming Relaxation [Held-Karp’72] 4 Integrality Gap:<br>
slide5. Previous Works Approximation Algorithms
log(n) [Frieze-Galbiati-Maffioli’82]
0.999 log(n) [Bläser’02]
0.842 log(n) [Kaplan-Lewenstein-Shafrir-Sviridenko’05]
0.666 log(n) [Feige-Singh’07]
O(logn/loglogn) [Asadpour-Goemans-Madry-O-Saberi’09]
O(1) (planar/bd genus) [O-Saberi’10,Erickson-Sidiropoulos’13]
Integrality Gap
≥ 2 [Charikar-Goemans-Karloff’06]
≤ O(logn/loglogn) [AGMOS’09]. 5<br>
slide6. Main Result 6 For any cost function, the integrality gap of the LP relaxation is polyloglog(n).<br>
slide7. Plan of the Talk 7 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys<br>
slide8. Thin Spanning Trees 8 Kn 2/n-thin tree Application: f(n)/k-thin trees imply 5f(n)-approx for ATSP<br>
slide9. Main Result 9 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide10. 10 In Pursuit of Thin Trees<br>
slide11. Main Ingredients 11 Interlacing Polynomials and
thin basis problem Effective resistance reduction
by Convex Programming<br>
slide12. A General Framework for L1 Opt Problems 12<br>
slide13. We can use convex duality to analyze Q(.) of the optimum. 13 Has exp. many constraints but can be solved using ellipsoid Since Q(.) is convex the program is convex<br>
slide14. 14<br>
slide15. 15<br>
slide16. Plan of the Talk 16 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys Min Cost
Flow Eff Resist
Reduction<br>
slide17. Thin Basis Prob [Marcus-Spielman-Srivastava’13] 17 d Linearly independent set of vectors<br>
slide18. A Weaker Suff Cond for Thin Basis Problem 18 ………………………………………<br>
slide19. Proof of Thin Basis Thm 19<br>
slide20. Method of Interlacing Polynomials 20<br>
slide21. Summary/Future Directions Main Ingredients:
Effective Resistance Reduction
Thin Basis Problem
Future Directions:
Algorithmic proof of [MSS’13].
Existence of C/k thin trees
Subsequent work: Svensson’s 27-app algorithm for graph ATSP. 21<br>
slide22. Previous Works: Randomized Rounding 22<br>
slide23. Main Result 23 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide24. Previous Works: Randomized Rounding 24<br>
slide25. Graph Laplacian 25 E.g.,<br>
slide26. A Necessary Condition for Spectral Thinness 26 where<br>
slide27. A k-con Graph with no Spectrally Thin Tree 27 n/k vertices k edges k edges<br>
slide28. A Sufficient Condition for Spectral Thinness 28<br>
slide29. Spectrally Thin Trees (Summary) 29 k-edge
connectivity O(1/k)-combinatorial thin tree O(1/k)-spectrally thin tree [MSS13] ?<br>
slide30. Our Approach 30<br>
slide31. Main Idea 31 Symmetrize L2 structure of G
while preserving its L1 structure<br>
slide32. An Example 32 n/k vertices<br>
slide33. An Observation 33<br>
slide34. Main Idea 34 D+G has a spectrally thin tree and
any spectrally thin tree of G+D is (comb) thin in G. Bypasses Spectral Thinness Barrier.<br>
slide35. An Impossibility Theorem 35<br>
slide36. Proof Overview 36 D is not a graph<br>
slide37. Proof Overview 37<br>
slide38. A Weaker Goal: Satisfying Degree Cuts 38<br>
slide39. A Convex Program for Optimum D 39<br>
slide40. Main Result 40 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
slide41. Conclusion Main Idea:
Symmetrize L2 structure of G while preserving its L1 structure
Tools:
Interlacing polynomials/Real Stable polynomials
Convex optimization
Graph partitioning
High dimensional geometry 41<br>
slide42. Future Works/Open Problems Algorithmic proof of [MSS’13] and our extension.
Existence of C/k thin trees and constant factor approximation algorithms for ATSP.
Subsequent work: Svensson designed a 27-app algorithm for ATSP when c(.,.) is a graph metric. 42<br>