On the Complexity ofTSP 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 TSPApp. [Chr76] Innap. [EK01]

Geometric TSPPTAS [Aro96,Mit96] NP-hard

[GGJ76,Pap77]

131

1303

2

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

G-TSP

Input:

Objective: Find the lightest tour hitting all

Ni

, , :G V E W E R

1 2, ... , m iN N N N V

G-TSPG-TSP is at least as hard as

TSPSet-Cover

Metric G-TSP Inapp. O(log n)(reduce from Hamilton

cycle) Geometric G-TSP

G-TSP in the PlaneApproximation 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] Constant Disjoint fat convex

G-TSP in the PlaneInapproximability Factors

Factor Type of Neighborhoods

[dB+02] Disjoint or Connected Regions(ESA02)

2041

2040

G-TSP in the PlaneMain Thm:[SaSc03]

Unless P=NP, G-TSP in the plane cannot be approximated to within

any constant factor.

Neighborhoods’ types and Inapproximability

Pairwise Disjoint Overlapping

Connected ? 2 -

Unconnected c c

G-TSP in the Plane

Neighborhoods’ types and Inapproximability

Pairwise Disjoint Overlapping

Connected c c

Unconnected 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

G-STInput:

Objective: Find the lightest tree hitting all Ni

Generalizes: Steiner-Tree Problem

Set-Cover Problem

, , :G V E W E R

1 2, ... , m iN N N N V

Most results for G-TSP hold for G-ST(Alg. & Inap., for various settings)constant approximation for G-TSPIffconstant approximation for G-STProof:

|Tree| ≤ |Tour| ≤ 2|Tree|

G-ST

Gap-Problems and Inapproximability

Minimization problem A

Gap-A-[syes, sno]

Gap-Problems and Inapproximability

Minimization problem A

Gap-A-[syes, sno]

Approximating A better than is NP-hard e

no

y s

S

S

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)

1

1,1Gap Ek VC

k

Ek-VC ≤p G-ST (on the plane)

H X = <G, W, N1,…,Nm>

n

1

Completeness

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

1-

2

n

1- n

Completeness

Proof:1

Soundness

3

1

n

k

t

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

1

n

k

Soundness

k

Proof:

k

3

12

2

n kT n

n n

kk k

1

2,

1

n

kGap G ST NP hard

n

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: D3

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

4

41

3k

4

2 21

3k

logO n

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.

1-1

log, is intractableGap Ek VC O

n

1

logd

dO n

Open Problems

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<pG-TSP on the plane

d

H = (V,E) G

From a vertex cover U to a natural Steiner tree TN(U)

|TN(U)| d|U| + 2

From a vertex cover U to a natural traversal TN(U)

|TN(U)| 2d|U| + 2

TSP

Gap-G-TSP-[1+ , 2 - ] is NP-hardGap-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