Download presentation

Presentation is loading. Please wait.

Published byDillon Harvel Modified over 2 years ago

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

2
TSP

3
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
Input: Objective: Find the lightest tour hitting all N i

8
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

9
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

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 constantfactor.

12
Neighborhoods’ types and Inapproximability Pairwise Disjoint Overlapping Connected ? 2 - Unconnected cc cc G-TSP in the Plane

13
Neighborhoods’ types and Inapproximability Pairwise Disjoint Overlapping Connected cc cc Unconnected cc cc 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
Input: Objective: Find the lightest tree hitting all N i Generalizes: Steiner-Tree Problem Set-Cover Problem

17
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

18
Gap-Problems and Inapproximability Minimization problem A Gap-A-[s yes, s no ]

19
Gap-Problems and Inapproximability Minimization problem A Gap-A-[ s yes, s no ] 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
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)

23
Ek-VC ≤ p G-ST (on the plane) H X = 1

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

25
Completeness Proof: 1

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
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

32
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< p G-TSP on the plane d H = (V,E) G

35
From a vertex cover U to a natural Steiner tree T N (U) |T N (U)| d|U| + 2

36
From a vertex cover U to a natural traversal T N (U) |T N (U)| 2d|U| + 2

37
TSP

38
Gap-G-TSP-[1+ , 2 - ] is NP-hard Gap-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

Similar presentations

Presentation is loading. Please wait....

OK

Approximation Algorithms for TSP

Approximation Algorithms for TSP

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on natural vegetation and wildlife of the world Ppt on e waste recycling Download ppt on live line maintenance techniques Ppt on opposites in hindi Ppt on lymphatic system Ppt on conference call etiquette in the workplace Ppt on marie curie Ppt on prime numbers Ppt on obesity prevention in children View my ppt online maker