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

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TSP

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

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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]

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

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G-TSP

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Input: Objective: Find the lightest tour hitting all N i

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

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G-TSP in the Plane Approximation Algorithms (Partial list) RatioType of Neighborhoods [AH94]Constantdisks, parallel segments of equal length, and translates of convex [MM95] [GL99] O(log n)Polygonal [DM01]ConstantConnected, comparable diameter [DM01]PTASDisjoint unit disks [dB + 02]ConstantDisjoint fat convex

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

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

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Neighborhoods’ types and Inapproximability Pairwise Disjoint Overlapping Connected ? 2 - Unconnected cc cc G-TSP in the Plane

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Neighborhoods’ types and Inapproximability Pairwise Disjoint Overlapping Connected cc cc Unconnected cc cc G-TSP in 3D G-TSP in the Plane

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

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G-ST

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Input: Objective: Find the lightest tree hitting all N i Generalizes: Steiner-Tree Problem Set-Cover Problem

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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| G-ST

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Gap-Problems and Inapproximability Minimization problem A Gap-A-[s yes, s no ]

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Gap-Problems and Inapproximability Minimization problem A Gap-A-[ s yes, s no ] Approximating A better than is NP-hard is NP-hard.

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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.

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

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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 Hyper-Graph Vertex-Cover (Ek-VC)

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Ek-VC ≤ p G-ST (on the plane) H X = 1

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

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Completeness Proof: 1

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

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Soundness Proof:

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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.

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

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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

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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. Open Problems

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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 ?

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THE END

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Hyper-Graph-Vertex-Cover< p G-TSP on the plane d H = (V,E) G

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From a vertex cover U to a natural Steiner tree T N (U) |T N (U)| d|U| + 2

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From a vertex cover U to a natural traversal T N (U) |T N (U)| 2d|U| + 2

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TSP

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

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

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Hyper-Graph Vertex-Cover

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