1. 1 On the Hardness Of TSP with Neighborhoods and related Problems (some slides borrowed from Dana Moshkovitz) O. Schwartz & S. Safra

2. 2 Desire: A Tour Around the World

3. 3 The Problem: Traveling Costs Money 1795$

4. 4 But I want to do so much

5. 5 The Group-TSP (G-TSP) A Minimal cost tour, but All goals are accomplished. TSP with Neighborhoods One of a Set TSP Errand Scheduling

6. 6 The G-TSP Generalizes: TSP Hitting Set

7. 7 G-TSP - The Euclidean Variant TSP – PTAS [Aro96, Mit96] Hitting Set – hardness factor log n [Fei98] Which is it more like ?

8. 8 Approximations [AH94] – Constant for well behaved regions. [MM95],[GL99] – O(log n) for more generalized cases. [DM01] – PTAS for unit disk. [dBGK + 02] – Constant for Convex fat objects.

9. 9 Group Steiner Tree (G-ST) Say you have a network, with links between some components, each with different capabilities (fast computing, printing, backup, internet access, etc). Each link can be protected against monitoring, at a different cost. The goal is to have all capabilities accessible through protected lines (at least for some nodes on the net).

10. 10 The G-ST A minimal cost tree, but All capabilities are accessible. Class Steiner Problem Tree Cover Problem One of a Set Steiner Problem

11. 11 The G-ST Generalizes: Steiner Tree - Each location contains a single distinguished goal. Hitting Set - The graph is complete and all edges are of weight 1.

12. 12 G-ST - The Euclidean Variant ST – PTAS [Aro96, Mit96] Hitting Set – hardness factor - log n [Fei98] Which is it more like ?

13. 13 Some Parameters of the Geometric Variant Dimension of the Domain Is each region connected ? Are regions Pairwise Disjoint ?

14. 14 Mitchell’s Open Problems [Mit00] [21] Is there an O(1)-approximation for the group Steiner problem on a set of points in the Euclidean plane ? [27] Does the TSP with connected neighborhoods problem have a polynomial-time O(1)-approximation algorithm ? What if neighborhoods are not connected sets (e.g. if neighborhoods are discrete sets of points) ? [30] Give an efficient approximation algorithm for watchman routes in polyhedral domain.

15. 15 Previous Result [dBGK + 02] G-TSP in the plane cannot be approximated to within unless P = NP Holds for connected sets, but not necessarily for pairwise disjoint sets.

16. 16 Our Results G-TSP and G-ST Dimension2-D3-Dd Pairwise Disjoint Sets YesNoYesNoYesNo Connected sets -2 -  Unconnected sets Resolving [Mit00, o.p. 21 and 27] dBGK+02 Improving [dBGK+02] And resolving [Mit00, o.p. 30] regarding WT & WP

17. 17 gap- G-TSP-[ a, b ] YES - There exists a solution of size at most b. NO - The size of every solution is at least a. Otherwise – Don’t care.

18. 18 From Gap to Inapproximability If we can show it’s NP-hard to distinguish between two far off cases, then it’s also hard to even approximate the solution. the size of the min-Traversal is extremely small the size of the min-Traversal is tremendously big Similarly for G-ST

19. 19 gap- G-TSP-[ a, b ] If gap- G-TSP-[ a, b ] is NP-hard then (for any  > 0) approximate G-TSP to within is NP-hard

20. 20 Gap Preserving Reductions Gap-VCGap-G-ST YES don’t care NO YES don’t care NO

21. 21 Hyper-Graphs A hyper-graph G=(V,E), is a set of vertices V and a set of edges E, where each edge is a subset of V. We call it a k-hyper- graph if each edge is of size k.

22. 22 VERTEX-COVER in Hyper-Graphs Instance: a hyper-graph G. Problem: find a set U  V of minimal size s.t. for any (v 1,…, v k )  E, at least one of the vertices v 1,…, v k is in U.

23. 23 How hard is Vertex Cover ? Theorems: z[Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[1- , k -19 ] is NP-hard z[DGKR02] Gap-k-hyper-graph-VC-[1- , (k-1-  ) -1 ] is NP-hard ( for k > 4 )  [DGKR02] Gap-hyper-graph-VC-[1- , O(log -1/3 n)] is intractable unless NP µ TIME (n O(log log n) )

24. 24 Main Result Thm: G-ST in the plane is hard to approximate to within any constant factor. Proof: By reduction from Gap-Hyper-Graph-Vertex-Cover. We’ll show that for any k, Gap-ST-[ ] is NP-hard

25. 25 The Construction: X 1

26. 26 Completeness Claim: If every vertex cover of G is of size at least (1-  )n then every solution T for X is of size at least (1-  )n-1. Proof: Trivial.

27. 27 Soundness Lemma: If there is a vertex cover of G of size at most then there is a solution T for X of size at most.

28. 28 Proof: A Natural Tree T N (U)

29. 29 Proof: A Natural Tree T N (U)

30. 30 Therefore, from the NP-hardness of [Tre01] Gap-k-hyper-graph-VC-[ ] we deduce that Gap-ST-[ ] is NP-hard Hence, (as k is arbitrary large), G-ST in the plane cannot be approximated to within any constant factor, unless P=NP. ▪

31. 31 Using A Stronger Complexity Assumption [DGKR02] Gap-hyper-graph-VC-[ ] is intractable unless NP µ TIME (n O(log log n) ) we deduce that Gap-ST-[ ] in the plane is intractable unless NP µ TIME (n O(log log n) ) Hence, G-ST in the plane cannot be approximated to within unless NP µ TIME (n O(log log n) ). ▪

32. 32 G-TSP Corollary 1: G-TSP cannot be approximated to within any constant factor unless P=NP. Corollary 2: G-TSP cannot be approximated to within unless NP µ TIME (n O(log log n) ).

33. 33 G-TSP Proof: any efficient  -approximation for G-TSP, yields an efficient 2  -approximation for G-ST (by removing an edge), as T * G-TSP · 2T * G-ST ▪

34. 34 How about log n ? Why not use the ln n hardness of [Fei98] ? (to obtain a factor of log ½ n)

35. 35 How hard is Vertex Cover ? Theorems: z[Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[1- , k -19 ] is NP-hard z[DGKR02] Gap-k-hyper-graph-VC-[1- , (k-1-  ) -1 ] is NP-hard ( for k > 4 )  [DGKR02] Gap-hyper-graph-VC-[1- , O(log -1/3 n)] is intractable unless NP µ TIME (n O(log log n) ) Completeness We need this (almost) perfect Completeness!

36. 36 Gap Location Theorems:  [Fei98] Gap-hyper-graph-VC-[t ln n, t ] is intractable unless NP µ TIME (n O(log log n) ) Where t<1 What’s the problem ?

37. 37 If the two properties are joint Conjecture: Gap-hyper-graph-VC-[1- , log -1 n ] is intractable unless NP µ TIME (n O(log log n) ) Corollary: G-TSP and G-ST cannot be approximated to within log ½ n, unless NP µ TIME (n O(log log n) )

38. 38 Other results Applying it to connected sets, dimension 3 and above. The case of sets of constant number of points. O(log 1/6 n) for Minimum Watchman Tour & Minimum Watchman Path. O(log 1/6 n) for Minimum Watchman Tour & Minimum Watchman Path. 2-  for G-TSP and G-ST with Connected sets in the plane. 2-  for G-TSP and G-ST with Connected sets in the plane. Dimension d – a hardness factor of and toward a factor of, which generalizes to. Open problems…

39. 39 Open Problems  Is Gap-hyper-graph-VC-[1- , log -1 n ] intractable unless NP µ TIME (nO(log log n)) ? zCan we do better than 2-  for connected sets in the plane ? Can we do anything for connected, pairwise disjoint sets on the plane ? zCan we avoid the square root loss ? zDoes higher dimension impel an increase in complexity ?

40. 40 2D unconnected to 3D connected

41. 41 Minimum Watchman Tour and Path

42. 42 Triangular Grid – For a better Constant 1

43. 43 G-TSP and G-ST – Connected sets in the plane Theorem: G-TSP and G-ST cannot be approximated to within 2- , unless P=NP Proof: By reduction from Hyper-Graph- Vertex-Cover.

44. 44 The construction d  F = E G = (V,E) G’

45. 45 The construction l

46. 46 Making it connected

47. 47 From a vertex cover U to a natural traversal T N (U) |T N (U)|  2d|U| + 2 

48. 48 From a vertex cover U to a natural Steiner tree T N (U) |T N (U)|  d|U| + 2 

49. 49 Natural is the Best Lemma: For some parameter d(  ), and for sufficiently large n and l, the shortest traversal (tree) is the natural traversal (tree) of a minimal vertex- cover.

50. 50 Natural is the Best

51. 51 Natural is the Best

52. 52 Natural is the Best

53. 53 Natural is the Best

54. 54 Natural is the Best

55. 55 Natural is the Best |T| ≥ |T’| ≥ |T N (U)|

56. 56 Natural is the Best

57. 57 Natural is the Best

58. 58 Natural is the Best |T| ≥ |T’| -  ≥ |T N (U)| - 

59. 59 Maximizing the Gap Ratio |T N (U YES )|  2d|U YES | + 2  |T N (U NO )|  2d|U NO | + 2  We want d as large as possible !

60. 60 Maximizing d – for G-ST D ≥ d  2( ρ+d )sin(  /n) ≥ d  2( ρ+d )  /n +  ≥ d  2  ρ/n +  ’ ≥ d ρ d D  /n |T N (U)|  2  |U|/n + 2 

61. 61 Maximizing d – for G-TSP D ≥ 2d +   2( ρ+d )sin(  /n) ≥ 2d +    ρ/n +  ’ ≥ d ρ d D  /n |T N (U)|  2  |U|/n + 2 

62. 62 G-TSP and G-ST in the Plane If Gap-k-Hyper-Graph-Vertex-Cover-[A,B] is NP- hard, then (for any  > 0) Gap-k-G-TSP-[1+A- ,1+B+  ] is NP-hard Gap-k-G-ST-[1+A- ,1+B+  ] is NP-hard

63. 63 G-TSP and G-ST in the Plane [Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[ 1- , k -19 ] is NP-hard Therefore (for any  > 0), Gap-G-TSP-[2- , 1+  ] is NP-hard and Gap-G-ST-[2- , 1+  ] is NP-hard. even if each set is connected ▪