Approximation Via Cost Sharing: Multicommodity Rent or Buy

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Approximation Via Cost Sharing Multicommodity
Rent or Buy

• Martin Pál
• Joint work with
• Anupam Gupta Amit Kumar
• Tim Roughgarden

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Steiner Forest and Rent or Buy
Input graph G, weights ce on edges, set of
demand pairs D, a constant M1
Solution a set of paths, one for each (si,ti)
pair. Goal minimize cost of the network built.
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The cost
Rent or buy Must rent or buy each edge.
rent pay ce for each path using edge e buy pay
M?ce Goal minimize rentalbuying costs.
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Related work

• 2-approximation for Steiner Forest Agarwal,
  Klein Ravi 91, Goemans Williamson 95
• Constant approximation for Single Sink Buy at
  Bulk Karger Minkoff 00, Meyerson
  Munagala 00
• Constant approximation for Multicommodity Rent or
  Buy Kumar, Gupta Roughgarden 02
• Spanning Tree Randomization Single Sink Rent
  or Buy Gupta, Kumar Roughgarden 03
• Cost sharing (Steiner tree, Single Sink Rent or
  Buy) Jain Vazirani 02, P Tardos 03

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Our work

• Simple randomized 8-approximation for the
  Multicommodity Rent or Buy problem
• Analysis based on cost sharing
• Ball inflation technique of independent interest
• Can be implemented in O(m log n) time

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The Algorithm
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Bounding the cost
Expected buying cost is small
Lemma There is a forest F on marked pairs with
Ecost(F) ? OPT.
e bought by OPT Pre is in F 1 e rented by
OPT Pre is in F 1/M
Using ?-approx, can get ? ? ? OPT Theorem
Expected cost of bought edges is ? ? ? OPT.
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Bounding the rental cost
Unmarked demand pays its rental cost Marked
demand contributes towards buying its tree
Plan Bound rental cost by the buying cost
Expected rental(?) ? ? Expected share(?)
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Need a cost sharing algorithm
Input Set of marked demands S Output Steiner
forest FS on S cost share ?(S,j) of each
pair j?S
Set of demands
demand pair
?(??, ?) 4 ?(??, ?) 5
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Cost Sharing for Steiner Forest
(P1) Constant approximation cost(FS) ? ?
St(S) (P2) Cost shares do not overpay ?j?S
?(S,j) ? St(S) (P3) Cost shares pay enough
let S S ? si, ti dist(si, ti) in G/ FS
? ? ?(S,i)
Example S ? i ?
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Bounding rental cost
Suppose we flip a coin for every demand pair
except ?. Let S be the set of marked pairs.
Erent(?) S ? dist(?,?) in G/FS
Erent(?) ? ? ? Ebuy(?)
? ? ?
Ebuy(?) S 1/M ? M ? ?(S?,?)
?j?D Erent(j) ? ? ?j?D Ebuy(j) ? ? OPT
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Computing shares using the Goemans-Williamson
algorithm
?(?)
?(?)
?(?)
Active terminals share cost of growth evenly.
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Goemans-Williamson is not enough
cost share 1/n ?rental 1 ?
Problem cost shares do not pay enough!
Solution Inflate the balls! Roughly speaking,
multiply each radius by ? gt 1
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Summing up
For ? 2 we get a 4-approximate 4-strict sharing
alg. This yields an 8-approximation algorithm in
total.
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Conclusions

• We have presented a simple 8-approx. for
  Multicommodity Rent or Buy.
• Extension instead of pairs, have groups. Each
  group must be connected by a tree, edge cost
  depends on the number of trees using it.
• Does our algorithm work with any Steiner forest
  algorithm? With unscaled Goemans-Williamson?
• Other cost sharing functions? (crossmonotonic..)

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What does (P3) say
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What does (P3) say
Solution for
Solution for
a
b
c
?( , ) ? ? (ac)
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Solution

• Inflate the balls !
• Run the standard AKR, GW algorithm
• Note the time Tj when each demand j deactivated
• Run the algorithm again, except that now every
  demand j deactivated at time ??Tj for some ? gt 1
• Adopting proof from GW buying cost at most
  2???OPT.

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How to prove (P3)
Need to compare runs on D and D D ? si,
ti Idea 1. pick a si, ti path P in G/ FD 2.
Show that ?(D,i) accounts for a constant
fraction of length(P)
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How to prove (P3)
Need to compare runs on D and D D ? si,
ti Idea 1. pick a si, ti path P in G/ FD 2.
Show that ?(D,i) accounts for a constant
fraction of length(P)
Instead of ?(D,i), use alone(i), the total time
si or ti was alone in its cluster
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How to prove (P3)
Need to compare runs on D and D D ? si,
ti Idea 1. pick a si, ti path P in G/ FD 2.
Show that ?(D,i) accounts for a constant
fraction of length(P)
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How to prove (P3)
Need to compare runs on D and D D ? si,
ti Idea 1. pick a si, ti path P in G/ FD 2.
Show that ?(D,i) accounts for a constant
fraction of length(P)
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How to prove (P3)
Need to compare runs on D and D D ? si,
ti Idea 1. pick a si, ti path P in G/ FD 2.
Show that ?(D,i) accounts for a constant
fraction of length(P)
?(D, ), 2.5 alone( ) 2.0
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Mapping of layers
Lonely layer generated by si or ti while the
only active demand in its cluster Each non-lonely
layer in the D run maps to ? layers in the
scaled D run. Idea length(P) of lonely and
non-lonely layers crossed in D run length(P) ?
of layers it crosses in scaled D run Hence for
each non-lonely layer, ?-1 lonely layers crossed.
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Mapping not one to one
Two non-lonely layers in D run can map to the
same layer in scaled D run.
D run
scaled D run
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Not all layers of D run cross P
A layer in D run that crosses P can map to a
layer in scaled D run that does not cross P
D run
scaled D run
si
si
The waste can be bounded.