Group strategy proof mechanisms via primaldual algorithms Cost Sharing

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Group strategy proof mechanisms via primal-dual
algorithms(Cost Sharing)

• Martin Pál Éva Tardos

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Our setting
Universe U of (selfish) users Users want to
benefit from shared infrastructure cost function
c()
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Sharing the cost
Known as core in game theory
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Modeling the users
ui value of service for i User may say NO if
pigtui util(i) ui pi if i gets service
0 otherwise
Cost sharing fn ? 2U?U ? ? ?(S,i) share of
user i, if set S served
?(U, ?) ?(U-?, ?)
cross-monotonic for i?S?T?U ?(S,i) ?(T,i)
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The MoulinShenker mechanism
S U repeat ask each user i is
?(S,i) ui ? drop all i?S who say NO until all
i?S say YES Output set S prices pi ?(S,i)
Theorem MoulinShenker ?(.) cross-monotonic
? mechanism group strategyproof
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Known cross-monotonic functions

• Exact cross-monotonic sharing exists if c()
  submodular
• MoulinShenker 98
• Exact cost sharing for spanning tree
• KentSkorin-Kapov 96, JainVazirani 01
• Any other games for which cross-mono sharing
  exists?

implies 2-approx. cost sharing for Steiner tree
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Our results
General method for generating competitive,
cross-monotonic cost shares using primal-dual
algorithms Used our method to construct a cost
sharing fn for
Metric Facility Location Single Sink Rent or Buy
recovers 1/3 of cost
recovers 1/15 of cost
competitive, cross-monotonic
constructive proof gives an approximation
algorithm
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Approximation algorithms

• Primal-dual approximation for Facility Location
• JainVazirani99 , Mahdian,YeZhang 02,
  MettuPlaxton 00
• Approximation for Single Sink Rent or Buy
  KargerMinkoff 00, MeyersonMunagala 00
• SwamyKumar 02, Gupta,KumarRoughgarden 03

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Single Sink Rent or Buy
U is a set of users. r is the sink (root)
node. graph G with edge lengths ce.
1) Find a path from each user to sink.
2) Rent or Buy each edge.
Rent pay ce for each path using e Buy pay M ?
ce Goal minimize rental buying cost.
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Facility Location
F is a set of facilities. U is a set of
users. cij is the distance between any i and j in
U ? F. (assume cij satisfies triangle inequality)
fi cost of facility i
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Facility Location
1) Pick a subset F of facilities to open
2) Assign every client to an open facility
Goal Minimize the sum of facility and assignment
costs Si?F fi Sj?S c(j,s(j))
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Existing facility location algorithms..
each user j raises its ?j ?j pays for
connection first, then for facility if facility
paid for, declared open (possibly cleanup phase
in the end)
?5
?5
?6
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...do not yield cross-monotonic shares
previously, ?(?)6
with ??, ?(?)8 ?? helped ? to stop earlier ?
failed to help ?
?3
?3
?8
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Ghost shares
After ?i freezes, continue growing its ghost
?i ghosts keep growing forever
?3
?3
?5.5
Fact Shares ?i are cross-monotonic. Pf more
users ? more ghosts ? facilities open sooner ?
?i can stop growing earlier
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Constructing a solution (1)
Sp set of users contributing to p at time of
opening contributor set tp time of opening
facility p
tp
Sp
p
facility p is well funded, if ?jtp/3 for every
j?Sp
Close down all facilities that are not well funded
Lemma For every facility p there is a nearby
well funded facility r s.t. dist(p,r) 2(tp- tr)
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Constructing a solution (2)
Problem user contributing to multiple
well-funded facilities Solution close all of
them but one (process by increasing tp)
tp
Sp
p
Lemma For every well funded facility p there is
a nearby open q such that dist(p,q) 2tp
tq
Sq
q
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Summing up
Every open facility is well funded, i.e. can be
paid for by Sp.
We do not lose much on assignment cost by closing
facilities.
Cost shares pay for 1/3 of the cost of the
solution.
Theorem There is a cross-monotonic cost sharing
function for facility location that recovers 1/3
of cost.
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Plan of attack for Rent or Buy

• Gather clients into groups
• (often done by a facility location algorithm)
• Build a Steiner tree on the gathering points

JainVazirani gave cost sharing fn for Steiner
tree Have cost sharing for facility location Why
not combine?
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One shot algorithm

• Generate gathering points and build a Steiner
  tree at the same time.
• Allow each user to contribute only to the least
  demanding (i.e. largest) cluster he is connected
  to.
• ? not clear if the shares can pay for the tree

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Conclusions
General method for generating competitive,
cross-monotonic cost shares using primal-dual
algorithms. Used our method to construct a cost
sharing fn forFacility Location Single Sink
Rent or Buy. Other problems admitting cross-mono
cost sharing? Steiner Forest? Covering
problems? Impossibility results?
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The End
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End.
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Cost shares
Each Steiner component C needs 1/second for
growth. Maintenance cost of C split among users
connected User connected to multiple components
pays only to largest component User connected to
root stops paying
?j ? fj(t) dt
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Growing ghosts
Grow a ball around every user uniformly When M or
more balls meet, declare gathering point Each
gathering point immediately starts growing a
Steiner component When two components meet, merge
into one
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Building a solution
Open all healthy well funded facilities Assign
each client to closest facility
Fact 1 for every facility p, there is an open
facility q within radius 2tp.
c(p,q) 2(tp-tq) c(q,q)2 tq
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Cost Sharing

• Internet many independent agents
• Not hostile, but selfish
• Willing to cooperate, if it helps them

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Steiner tree how to split cost
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Steiner tree how to split cost
OPT( ) 3
OPT( ) 5
p( ) 5-3 2
No fair cost allocation exists!
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Desirable properties of mechanisms

• Sj?S pjc(S)
• (budget balance)
• Only people in S pay
• (voluntary participation)
• No cheating, even in groups
• (group strategyproofness)
• If uj high enough, j guaranteed to be in S
• (consumer sovereignity)

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Cost sharing function

• ? 2U?U ? R
• ?(S,j) cost share of user j, given set S
• Competitiveness Sj?S ?(S,j) c(S)
• Cost recovery c(S)/ß Sj?S ?(S,j)
• Voluntary particpiation ?(S,j) 0 if j?S
• Cross-monotonicity for j?S?T
• ?(S,j) ?(T,j)

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Cost recovery
To prove cost recovery, we must build a
network. Steiner tree on all centers would be too
expensive ? select only some of the centers like
we did for facility location. Need to show how to
pay for the tree constructed.
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Paying for the tree
We selected a subset of clusters so that every
user pays only to one cluster. But users were
free to chose to contribute to the largest
cluster may not be paying enough. Solution use
cost share at time t to pay contribution at time
3t.
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The last slide

• x-mono cost sharing known only for 3 problems so
  far
• Do other problems admit cross-mono cost sharing?
• Covering problems? Steiner Forest?
• Negative result SetCover no better than O(n)
  approx
• Applications of cost sharing to design of
  approximation algorithms.