Minimum Back-Walk-Free Latency Problem

Yaw-Ling Lin

Dept Computer Sci. & Info. Management,Providence University, Taichung, Taiwan.

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Minimum Latency Problem (MLP)

• Starts from s, sending goods to all other nodes.

• Traveling Salesperson Problem (TSP): Server oriented

• MLP: Client oriented

• MLP is also known as repairman problem or traveling repairman problem (TRP)

s

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MLP: Formal Definition

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MLP vs. TSP• TSP: minimizes the salesman’s total time. Server oriented, egoistic.

– No contstant approximation algorithm for general case.– Christofides (1976): 3/2-approximation ratio for metric case; Arora (1992):

metric TSP does not have PTAS unless P=NP.– Arora (1998 JACM): PTAS on Euclidean case.

• MLP: minimizes the customers’ total time. Clients oriented, altruistic.– Alias: deliveryman problem, traveling repairman problem (TRP).– Afrati (1986): MAX-SNP-hard for metric case.– Goeman (1996): 10.78-approximation ratio for metric case (with Garg, 19

96FOCS, technique); 3.59-approximation ratio for trees.– Arora (1999 STOC): quasi-polynomial ( O(nO(log n) ) approximation scheme

for trees and Euclidean space. – Sitters (2002, IPCO): MLP on trees is NP-complete; not known for caterpi

llars.

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MBLP: Back-Walk Free

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An Example

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Our Results

• MBLP, given a starting point of G– Trees : O(n log n ) time– k-path : O(n log k) ; path is O(n) time– DAG : NP-Hard (Reduce from 3-SAT)

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Properties of MBLP on Trees

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Properties (contd’)

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Properties (3)

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Algo MBLP-Tree: Example

5

3 10

8 11 2

Select 11

5

14/2 10

8 2

Select 10

15/2

14/2

8 2

Select 8

15/2

22/3 2

Select and output 15/222/3 2

Select and output 22/3

2Select and output 2

Result : 5,10,3,11,8,2

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Algorithm MBLP-Tree

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Analysis of MBLP-Tree

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Properties of MBLP on k-Path

<4,2,3,8> is right-skew; < 5, 3, 4, 1, 2, 6 > is not.<5> <3,4> <1,2,6> is decreasing right-skew partitioned.

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Properties of k-Path (contd’)

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Path-Partition: Example

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Algorithm Path-Partition

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Algorithm k-Path

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Analysis of k-Path

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MBLP on DAG is NP-Complete

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NP-Completeness: Construction

-- Reduction from 3SAT: n literals:

S

x1 x1

x2 x2

x3 x3

x4 x4

… n literals

1 literal

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NP-Completeness: Construction

-- Reduction from 3SAT: k clauses:

0 0 0

0 0 0

… k clauses

1 caluse

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NP-Completeness: illustration

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NP-Complete Proof

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Conclusion

• MBLP is easier than MLP, at least on trees.• MBLP remains hard even on dag.• The idea of atomic subtours helps in finding effici

ent algorithms of MBLP on trees.• The idea of atomic sequence becomes right-skew

partition, implying the linear time algorithm on paths.

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Future Research

• MLP on caterpillars.

• MBLP: finding the good starting points on paths, trees.

• MBLP: multiple servers on trees, paths.