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

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 App.[Chr76] Innap. [EK01] Geometric TSP PTAS [Aro96,Mit96] NP-hard [GGJ76,Pap77]

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

G-TSP

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

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

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

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

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

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

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

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

G-ST

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

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

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

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

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.

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

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)

Ek-VC ≤ p G-ST (on the plane) H  X = 1

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

Completeness Proof: 1

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

Soundness Proof:

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. 

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

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

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

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 ?

THE END

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

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

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

TSP

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

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

Hyper-Graph Vertex-Cover