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On the Complexity of TSP with Neighborhoods and Related problems

On the Complexity of TSP with Neighborhoods and Related problems. Muli Safra & Oded Schwartz. TSP. TSP. Input: G = (V,E) , W : E  R + Objective: Find the lightest Hamilton-cycle. TSP. TSP NP-Hard Even to approximate (reduce from Hamilton cycle) Metric TSP

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On the Complexity of TSP with Neighborhoods and Related problems

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  1. On the Complexity of TSP with Neighborhoods and Related problems Muli Safra & Oded Schwartz

  2. TSP

  3. TSP Input: G = (V,E) , W : E  R+ Objective: Find the lightest Hamilton-cycle

  4. TSP • TSP NP-Hard Even to approximate (reduce from Hamilton cycle) • Metric TSP App. [Chr76] Innap. [EK01] • Geometric TSP PTAS [Aro96,Mit96] NP-hard [GGJ76,Pap77]

  5. G-TSP AKA: • Group-TSP • Generalized-TSP • TSP with Neighborhoods • One of a Set TSP • Errand Scheduling • Multiple Choice TSP • Covering Salesman Problem

  6. G-TSP

  7. G-TSP Input: Objective: Find the lightest tour hitting all Ni

  8. G-TSP G-TSP is at least as hard as TSP Set-Cover • Metric G-TSP Inapp. O(log n) (reduce from Hamilton cycle) • Geometric G-TSP

  9. G-TSP in the Plane Approximation Algorithms (Partial list) Ratio Type of Neighborhoods [AH94] Constant disks, parallel segments of equal length, and translates of convex [MM95] [GL99] O(log n) Polygonal [DM01] Constant Connected, comparable diameter [DM01] PTAS Disjoint unit disks [dB+02] ConstantDisjoint fat convex

  10. G-TSP in the Plane Inapproximability Factors Factor Type of Neighborhoods [dB+02] Disjoint or Connected Regions (ESA02)

  11. G-TSP in the Plane Main Thm:[SaSc03] Unless P=NP, G-TSP in the plane cannot be approximated to within any constant factor.

  12. G-TSP in the Plane

  13. G-TSP in 3D G-TSP in the Plane

  14. G-ST AKA: • Group Steiner Tree Problem • Class Steiner Tree Problem • Tree Cover Problem • One of a Set Steiner Problem

  15. G-ST

  16. G-ST Input: Objective: Find the lightest tree hitting all Ni Generalizes: Steiner-Tree Problem Set-Cover Problem

  17. G-ST Most results for G-TSP hold for G-ST (Alg. & Inap., for various settings) constant approximation for G-TSP Iff constant approximation for G-ST Proof: |Tree| ≤ |Tour| ≤ 2|Tree|

  18. Gap-Problems and Inapproximability Minimization problem A Gap-A-[syes,sno]

  19. Gap-Problems and Inapproximability Minimization problem A Gap-A-[syes,sno]  Approximating A better than is NP-hard is NP-hard.

  20. Gap-Problems and Inapproximability Thm: [SaSc03] Gap-G-ST-[o(n),(n)] is NP-hard.  G-ST is NP-hard to approximate to within any constant factor. So is G-TSP in the plane.

  21. Hyper-Graph Vertex-Cover (Ek-VC) Input: H = (V,E) - k-Uniform-Hyper-Graph Objective: Find a Vertex-Cover of Minimal Size

  22. Hyper-Graph Vertex-Cover (Ek-VC) Input: H = (V,E) - k-Uniform-Hyper-Graph Objective: Find a Vertex-Cover of Minimal Size Thm:[D+02] For k>4 is NP-Hard

  23. 1 Ek-VC ≤p G-ST (on the plane) • H  X = <G, W, N1,…,Nm>

  24. Completeness Claim: If vertex-cover of H is of size then  tree cover T for X is of size

  25. 1 Completeness Proof:

  26. Soundness Claim: If  vertex cover of H of size then  tree cover T for X is of size

  27. Soundness Proof:

  28. Gap-G-ST (on the plane) k may be arbitrary large Unless P = NP, G-ST in the plane cannot be approximated to within any constant factor. 

  29. Problem Variants Variants: 2D unconnected, overlapping (G-ST & G-TSP) unconnected, pairwise-disjoint Variants: D3 Holds for connected variants too.

  30. Other Corollaries Small sets size: k-G-TSP in the plane k-G-ST in the Plane Watchman Tour and Watchman path problems in 3D cannot be approximated to within any constant, unless P=NP

  31. Open Problems If the two properties are joint: then Approximating G-TSP and G-ST in the plane to within is intractable. Approximating G-TSP and G-ST in dimension d within is intractable.

  32. Open Problems Is 2 the approximation threshold for connected overlapping neighborhoods ? Is there a PTAS for connected, pairwise disjoint neighborhoods ? How about watchman tour and path in the plane ? Does any embedding in the plane cause at least a square root loss ? Does higher dimension impel an increase in complexity ?

  33. THE END

  34. Hyper-Graph-Vertex-Cover<pG-TSP on the plane G H = (V,E) d 

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

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

  37. TSP

  38. Gap-G-TSP-[1+ ,2 -  ] is NP-hardGap-G-ST-[1+ , 2 -  ] is NP-hard How to connect it ?

  39. Neighborhood TSP and ST– - Making it continuous How about the unconnected variant ?

  40. Hyper-Graph Vertex-Cover

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