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 - (O(m log
  n) running time)
• Analysis based on strict cost sharing - useful
  in other contexts with two-stage costs
  (stochastic opt.)
• Achieving strictness Ball inflation technique

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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) Good approximation cost(FS) ? ?
St(S) (P2) Cost shares do not overpay ?j?S
?(S,j) ? St(S) (P3) Strictness 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 AKR-GW algorithm
?(?)
?(?)
?(?)
Active terminals share cost of growth evenly.
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AKR-GW is not enough
cost share(?) 1/n ?rental(?) 1 ?
Problem cost shares do not pay enough!
Solution Force the algorithm to buy the middle
edge - need to be careful not to pay too much
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Forcing AKR-GW buy more
Idea Inflate the balls! Roughly speaking,
multiply each radius by ? gt 1
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Why should inflating work?
layer only ? is contributing layer terminals
other than ? contributing
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The inflated AKR-GW

• Run the standard AKR-GW algorithm on S
• Note the time Tj when each demand j frozen
• Run the algorithm again, with new freezing
  ruleevery demand j deactivated at time ??Tj for
  some ? gt 1

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The inflated AKR-GW (2)
Freezing times
Original AKR-GW
Inflated AKR-GW
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Easy facts
Fact Any time t u and v in the same cluster in
Original ? u and v in the same cluster
in Inflated ? Inflated has at most as many
clusters as Original
Theorem Forest constructed by Inflated AKR-GW
has cost at most 2??OPT. Pf adopted from GW.
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Its Hammertime Proving strictness

• Compare
• Original(S?) (cost shares)
• Inflated(S) (the forest we buy)
• Need to prove dist(s?, t?) in G/ FS ? ? ?(S?,
  ?)
• Lower bound on ?(S?, ?) alone(s?) alone(t?)
• Original(S?) must have connected ? terminals.
  Hence it bought a s? -- t? path. Use it to bound
  dist(s?, t?).

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Simplifying..

• Let ?? be time when Original(S?) connects ?
  terminals.
• Terminate Original(S?) at time ??.
• In Inflated(S), freeze each term. j at time
  ??min(Tj, ??).
• Simpler graph H
• Contract all edges that Original(S?) bought.
  Call the new graph H.
• Run both Original(S?) and Inflated(S) on H.
• Note that Inflated(S) on H does not buy any edges!

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Comparing Original(S?) Inflated(S)
Freezing times
Original(S?)
Inflated(S)
Freezing times
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Proof idea
Correspondence of other (i.e. non-red)
layers Layer l in Original(S?) ? Layer l in
Inflated(S)
P Alone Other ? ? Other
P
?Waste
?Waste
If we can prove Waste Alone, we are done.
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Bounding the Waste
Claim If a layer l contains s?, then its
corresponding inflated layer l also contains s?.
Same for t?.
Pf By picture.
Original(S?)
Inflated(S)
The only way of wasting a layer l is when l does
not contain s? (or t?), but l does.
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Bounding the Waste (2)
l
l
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Bounding the Waste (3)
l
l
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Summing up
P 2?/(?-1) ? Alone Hence we have 2?
approximation algorithm admitting a
2?/(?-1)-strict cost sharing. Setting ?2 we
obtain a 448 approximation.
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Future work

• Does the sampling algorithm work with any Steiner
  forest algorithm? With unscaled AKR-GW?
• Does group strictness hold? augment(sol(S),
  T) ? ???j?T ?(S,i)
• Derandomize the sampling algorithm.
• Other applications of cost sharing and
  strictness.
• Different cost sharing functions?
  (crossmonotonic..)

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Useful Garbage
Theorem Inflated AKR-GW is a 2?
approximation. Pf adopted from GW.
????v??