Minimum Back-Walk-Free Latency Problem Yaw-Ling Lin Dept Computer Sci. & Info. Management, Providence University, Taichung, Taiwan.

Yaw-Ling Lin, Providence, Taiwan2 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

Yaw-Ling Lin, Providence, Taiwan3 MLP: Formal Definition

Yaw-Ling Lin, Providence, Taiwan4 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): approximation ratio for metric case (with Garg, 1996FOCS, technique); 3.59-approximation ratio for trees. –Arora (1999 STOC): quasi-polynomial ( O(n O(log n) ) approximation scheme for trees and Euclidean space. –Sitters (2002, IPCO): MLP on trees is NP-complete; not known for caterpillars.

Yaw-Ling Lin, Providence, Taiwan5 MBLP: Back-Walk Free

Yaw-Ling Lin, Providence, Taiwan6 An Example

Yaw-Ling Lin, Providence, Taiwan7 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)

Yaw-Ling Lin, Providence, Taiwan8 Properties of MBLP on Trees

Yaw-Ling Lin, Providence, Taiwan9 Properties (contd’)

Yaw-Ling Lin, Providence, Taiwan10 Properties (3)

Yaw-Ling Lin, Providence, Taiwan11 Algo MBLP-Tree: Example Select / Select 10 15/2 14/2 82 Select 8 15/2 22/3 2 Select and output 15/2 22/3 2 Select and output 22/3 2 Select and output 2 Result : 5,10,3,11,8,2

Yaw-Ling Lin, Providence, Taiwan12 Algorithm MBLP-Tree

Yaw-Ling Lin, Providence, Taiwan13 Analysis of MBLP-Tree

Yaw-Ling Lin, Providence, Taiwan14 Properties of MBLP on k-Path is right-skew; is not. is decreasing right-skew partitioned.

Yaw-Ling Lin, Providence, Taiwan15 Properties of k-Path (contd’)

Yaw-Ling Lin, Providence, Taiwan16 Path-Partition: Example

Yaw-Ling Lin, Providence, Taiwan17 Algorithm Path-Partition

Yaw-Ling Lin, Providence, Taiwan18 Algorithm k-Path

Yaw-Ling Lin, Providence, Taiwan19 Analysis of k-Path

Yaw-Ling Lin, Providence, Taiwan20 MBLP on DAG is NP-Complete

Yaw-Ling Lin, Providence, Taiwan21 NP-Completeness: Construction -- Reduction from 3SAT: n literals: Sx1x1 x1x1  x2x2 x2x2  x3x3 x3x3  x4x4 x4x4  … n literals 1 literal

Yaw-Ling Lin, Providence, Taiwan22 NP-Completeness: Construction -- Reduction from 3SAT: k clauses:  000  000 … k clauses 1 caluse

Yaw-Ling Lin, Providence, Taiwan23 NP-Completeness: illustration

Yaw-Ling Lin, Providence, Taiwan24 NP-Complete Proof

Yaw-Ling Lin, Providence, Taiwan25 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.

Yaw-Ling Lin, Providence, Taiwan26 Future Research MLP on caterpillars. MBLP: finding the good starting points on paths, trees. MBLP: multiple servers on trees, paths.