Exploring TSP and Related Problems with Neighborhoods

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

2. Desire: A Tour Around the World

3. The Problem: Traveling Costs Money 1795$

4. But I want to do so much

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

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

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

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

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. 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. Our Results Improving [dBGK+02] And resolving [Mit00, o.p. 30] regarding WT & WP Resolving [Mit00, o.p. 21 and 27]

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. 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. 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. Gap Preserving Reductions Gap-VC Gap-G-ST • YES • YES • don’t care • don’t care • NO • NO

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

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. The Construction: X 1

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. 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. Proof: A Natural Tree TN(U)

29. Proof: A Natural Tree TN(U)

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

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 (nO(log log n)).

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. How about log n ? Why not use the ln n hardness of [Fei98] ? (to obtain a factor of log½n)

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

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

37. If the two properties are joint Conjecture: Gap-hyper-graph-VC-[1- ,log-1n] is intractable unless NP µ TIME (nO(log log n)) Corollary: G-TSP and G-ST cannot be approximated to within log½n, unless NP µ TIME (nO(log log n))

38. Other results • Applying it to connected sets, dimension 3 and above. • The case of sets of constant number of points. • O(log1/6 n) for Minimum Watchman Tour & Minimum Watchman Path. • 2- for G-TSP and G-ST with Connected sets in the plane. • Dimension d – a hardness factor ofand toward a factor of , which generalizes to . Open problems…

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

40. 2D unconnected to 3D connected

41. Minimum Watchman Tour and Path

42. Triangular Grid – For a better Constant 1

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

45. The construction l

46. Making it connected

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

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

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. Natural is the Best