Paths, Trees and Minimum Latency Tours

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Paths, Trees and Minimum Latency Tours

• Kamalika Chaudhuri,
• Brighten Godfrey,
• Satish Rao,
• Kunal Talwar
• UC Berkeley

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The Problem

• Given
• V Set of points
• d Distance function on pairs of points
• s Starting point
• Find a tour of all points, starting at s, which
  minimizes the total latency
• Also called the Traveling Repairman problem

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Some Results

• SG74 NP-Hard on general graphs
• Sitters02 NP Hard on weighted trees
• BCCPRS94 MAX-SNP Hard on general graphs
• BCCPRS94 Constant factor algorithm for metric
  spaces
• GK96 Approximation ratio 7.18 e
• Our approximation ratio 3.59

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An Algorithm BCCPRS94

• For j1,2,3,..
• Find a tree Tj of cost at most 2j which has the
  most vertices ()
• Double Tj and shortcut to get tour Pj
• Concatenate tours P1,P2,

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Analysis

• Suppose 2j latencyOPT(i) 2j1
• Tj1 has at least i vertices
• Latency of the ith vertex in our tour is at most
• 2 2j1 2 ?k 2k
• 8 latencyOPT(i)
• Problem
• Assumed that we can find exact solution to k-MST
  (the minimum spanning tree with k vertices)
• ? approximate k-MSTs approximation factor 8 ?

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Trees vs. Paths

• BCCPRS94, GK96
• Lower bound k-MST
• Tours from k-MSTs
• Tours of geometrically increasing lengths
• 3.59 ? ¼ 7.18 approximation

• Our Algorithm
• Lower bound k-stroll
• Tours from good k-trees
• Tours of geometrically increasing lengths
• 3.59 approximation

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Trees vs. Paths

• Our Algorithm
• Lower bound k-stroll
• Tours from good k-trees
• Tours of geometrically increasing lengths
• 3.59 Approximation

• This talk
• k-(stroll, tree)
• Finding good k-trees

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Paths vs. Trees

• Our contribution
• Use k-stroll as a lower bound instead of k-MST

k-stroll Given s, the minimum cost path from
s with k vertices But k-stroll does not
seem any easier than k-MST !
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Good k-trees

• Good k-tree
• k vertices
• Tree cost optimal k-stroll
• Find a good k-tree by a modification of the k-MST
  algorithm Garg96,AK00

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Finding good k-trees

• Garg96, AK00 use a variant of the primal-dual
  algorithm of GW92
• Allot a budget ? to each vertex
• Different ? s produce trees of different size k?

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Finding good trees

• Our algorithm
• Fix endpoint t
• Budget 8 to t, ? to all other vertices
• Run the primal dual algorithm
• This may not give trees for all k
• Use Garg96,AK00 to find trees for all k
• Argue ALW02 that we need only the trees
  produced

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Analysis Basic Ideas
Tree LP min ?e ce xe ?e 2 ?(S)xe 1 8
S ½ V
Path LP min ?e ce xe ?e 2 ?(S) xe 1
8 S ½ V s, t 2 S ?e 2 ?(S) xe 2 8 S ½ V
s,t
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Analysis Dual LPs

• Tree LP
• max ?S yS
• ?Se 2 ?(S)yS ce 8 e

• Path LP
• max 2?S yS - ?Tt 2 T yT
• ?Se 2 ?(S)yS ce 8 e

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Analysis Dual LPs

• Tree LP
• max ?S yS
• ?Se 2 ?(S)yS ce 8 e

• Path LP
• max 2?S yS - ?Tt 2 T yT
• ?Se 2 ?(S)yS ce 8 e

Tree Primal Cost
Tree Dual Cost
Path Dual Cost
¼
2(1-1/n)

GW92
) Cost of the tree Cost of Opt Path
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Analysis Basic Ideas
Tree LP min ?e ce xe ?e 2 ?(S)xe 1 8 S
½ V
Path LP min ?e ce xe ?e 2 ?(S) xe 1 8
S ½ V s, t 2 S ?e 2 ?(S) xe 2 8 S ½ V
s,t
Tree Primal Cost
Tree Dual Cost
Path Dual Cost
¼
2(1-1/n)
GW92
) Cost of the tree Cost of Opt Path
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Running Time

• Running Time O(n3 log n)
• O(n2) time to run primal dual
• O(log n) values of ?
• O(n) guesses for t
• Example shows guessing t appears to be necessary

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Conclusion

• Improved approximation factors for
• Minimum latency 3.59
• k-Minimum latency 8.47
• GK96 3.59 is the best we can do by stitching
  together tours
• Is there an LP based approach which does better?
• FLT02 Better approximation for minimum latency
  set cover