Minimum Back-Walk-Free Latency Problem

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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)

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MLP Formal Definition
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MLP vs. TSP

• TSP minimizes the salesmans 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 PNP.
• 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, 1996FOCS, 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 caterpillars.

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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
Select 11
Select 10
Select 8
Result 5,10,3,11,8,2
Select and output 15/2
Select and output 2
Select and output 22/3
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
lt4,2,3,8gt is right-skew
lt 5, 3, 4, 1, 2, 6 gt is not.
lt5gt lt3,4gt lt1,2,6gt 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
n literals
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NP-Completeness Construction
-- Reduction from 3SAT k clauses
k clauses
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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
  efficient 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.