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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