1. Approximating ATSP by Relaxing Connectivity Ola Svensson

2. Best ticket from Lausanne to Paris Lausanne Leaving from:Departing: 04/06/2016 Paris Going to: Returning: 08/06/2016 Search for flights

3. Best ticket from Lausanne to Paris Patience (algorithm working)

4. Best ticket from Lausanne to Paris Ticket found

5. Best ticket from Lausanne to Paris What is the shortest way to visit these cities?

6. Best ticket from Lausanne to Paris What is the shortest way to visit these cities?

7. Best ticket from Lausanne to Paris What is the shortest way to visit these cities? Traveling Salesman Problem

8. Variants studied in mathematics by Hamilton and Kirkman already in the 1800’s Benchmark problem in computer science from the “beginning” Today, probably the most studied NP-hard optimization problem Intractable: (current) exact algorithms require exponential time Major open problem what efficient computation can accomplish

9. Two Basic Versions

10. SYMMETRIC: distance from u to v equals distance from v to u u u v v 3 3

11. Two Basic Versions SYMMETRIC: distance from u to v equals distance from v to u ASYMMETRIC: more general and no such assumption is made u u v v 3 3 u u v v 3 100

12. Motivation of Asymmetric TSP vs

13. Asymmetric Traveling Salesman Problem

14. 3 1 1 5000

15. Asymmetric Traveling Salesman Problem 3 1 1 5000 in-degree = out-degree

16. Asymmetric Traveling Salesman Problem Held-Karp Relaxation

17. QUESTION: What’s the strength of standard LP?

18. Previously Two Approaches Held-Karp Relaxation Easy to Find Eulerian graphEasy to Find Connected Graph

19. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

20. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

21. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

22. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

23. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

24. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

25. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected

26. Repeated Cycle Cover Find min-cost cycle cover “Contract“ Repeat until graph is connected Frieze, Galbiati, Maffiolo’82

27. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

28. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

29. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

30. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

31. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

32. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

33. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

34. Spanning Tree Approach Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge

35. The tree needs to be thin! Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10 (follows by the Hoffman’s circulation theorem)

36. State of the Art of Thin Tree Approach Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10 Anari, Oveis Gharan’14

37. To summarize the two approaches This is believed to be close to the truth No better guarantees for shortest path metrics on unweighted graphs for which there was recent improvements for the symmetric TSP Oveis Gharan, Saberi & Singh Mömke & S Mucha Sebö & Vygen

38. To summarize the two approaches This is believed to be close to the truth No better guarantees for shortest path metrics on unweighted graphs for which there was recent improvements for the symmetric TSP Main difficulty in cycle cover approach is to bound #iterations In thin-tree approach we reduce ATSP to an unweighted problem

39. A NEW APPROACH Find Eulerian graph with some connectivity requirements

40. Relaxing Connectivity Instead of Do for smart C

41. Our result THEOREM: For ATSP on shortest path metrics, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm.

42. Our result THEOREM: For ATSP on shortest path metrics, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm. 2

43. Our result THEOREM: For ATSP on shortest path metrics, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm. 2 2 1 1 1+1+2+2=6

44. Our result THEOREM: For ATSP on shortest path metrics, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm. Result also generalizes to node-weighted metrics 2 2 1 1 1+1+2+2=6

45. OUR APPROACH Cousin to repeated cycle cover approach Distant relative to thin tree approach

46. Relaxing Connectivity Instead of Do for smart C

47. Local Connectivity ATSP

50. 1

51. 1 Not hard to generalize to general metrics Please see paper

52. Our main technical result

53. The problems are equivalent up to a small constant factor There is an easy 3-light algorithm for node-weighted metric (only part where special metric is used)

54. PROOF IDEA OF MAIN THM

55. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

56. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

57. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

58. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

59. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

60. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

61. Repeatedly solve Local-Connectivity ATSP with the current connected subgraphs as partitions

62. Lexicographic Initialization

64. F

65. F

66. F

68. …

69. Merging Step

76. We show that it is always possible to connect the graph by charging each initialization component a constant number of times which gives main result

77. Some open problems…

78. Node-Weighted Symmetric TSP 1 1 3 3 1 1 3 3 2 2 3 5 5 4 4 4 Can you do better than Christofides (1.5)?

79. Constant for ATSP on General Metrics?

80. Summary