12 Sep 03 Martin Pál: Cost sharing & Approx 1

Cost Sharing and Approximation

Martin Pál

joint work with Éva Tardos

A-exam

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

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How much does it pay to share?

# users

cost

No sharing

Steiner treecost

# users

# users

costM Rent or buy

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Steiner tree: how to split cost

4 432 2 ?

? ?1

3 30

4 33

2 23

4 4

Cost shares Cost shares

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OPT( ) = 5p( ) = 5-3 = 2

OPT( ) = 3

Steiner tree: how to split cost

No “fair” cost allocation exists!

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Utilities

$45u1=25

u2=25

u3=25

internetp1=15

p2=15

p3=15

uj – utility of agent j

u1’=5

Problem: users may not reveal true utilities

p1=0

p2=22.5p3=22.5

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Cost sharing mechanism..is a protocol/algorithm that

• requests utility uj from every user j• selects the set S of people serviced• builds network servicing S• Computes the payment pj of every user

jS

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Desirable properties of mech’s• ΣjS pj=c*(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)

ΣjS pj ≤ c*(S)(competitiveness)ΣjS pj ≥ c(S)/β(approx. cost recovery)

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Cost sharing functionξ : 2UU ℛ

ξ(S,j) – cost share of user j, given set S

• Competitiveness: ΣjS ξ(S,j) ≤ c*(S)• Cost recovery: c(S)/β ≤ ΣjS ξ(S,j)• Voluntary particpiation: ξ(S,j) = 0 if jS• Cross-monotonicity: for jST

ξ(S,j) ≥ ξ(T,j)

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The Moulin&Shenker mechanismLet ξ : 2UU ℛ be a cost sharing function.S Uwhile unhappy users exist

offer each jS service at price ξ(S,j)

S S – {users j who rejected}output the set S and prices pj = ξ(S,j)Thm: [Moulin&Shenker]: ξ(.) cross-monotonic mech. group strategyproof

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Designing x-mono functionsWe construct cross-monotonic cost shares for two games:•Metric facility location game•Single source rent or buy gameFacility location: competitive, recovers 1/3 of costRent or buy:competitive, recovers 1/15 of cost

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Facility LocationF is a set of facilities.D is a set of clients.cij is the distance between any i and j in D

F.(assume cij satisfies triangle inequality)

fi: cost of facility i

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Facility Location1) Pick a subset F’ of facilities to open2) Assign every client to an open facilityGoal: Minimize the sum of facility and assignment costs: ΣiF’ fi + ΣjS c(j,σ(j))

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Existing 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)

=4

=4

=6

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..do not yield x-mono shares

=5

=5

with , ( )=6without , ( )=5 was stopped prematurely in the first run

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Ghost shares

=4

=4

=5

Two shares per user:•ghost share j

•real share jj grows foreverj stops when connected

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Easy factsFact 1: cost shares j are cross-monotonic.Pf: More users opens facilities faster each j can only stop growing earlier. Fact 2 [competitiveness]: ΣjS j ≤ c*(S).Pf: j is a feasible LP dual.

Hard part: cost recovery.

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Constructing a solutiontp: time when facility p openedSp: set of clients connected to p at time tp

facility p is well funded, if 3j≥tp for every jSp

each facility is either poisoned or healthyp is poisoned if it shares a client with a well funded healthy facility q and tp > tq

=7

=2

=2

tp=2

tq =7

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Building a solutionOpen 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.

≤tp

j≤tp/3tp1 ≤ j

≤tp1tp

p≤tp1 ...

qtq

Well funded

q’healthyc(p,q) ≤2(tp-tq)

c(q,q’)≤2 tq c(p,q’) ≤tp

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Cost recoveryFact 2: p open clients in Sp can pay 1/3 their connection + facility costPf: fj = ΣjS(p) tp – cjp and j≥tp/3

Sp

Fact 3: j is in no Sp can pay for 1/3 of connectionPf: fj = ΣjS(p) tp – cjp and j≥tp/3

open

≤j

≤2j

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Cost recoverySummary:•Cost shares can pay for 1/3 of soln we construct•Never pay more than cost of the optimum•With increasing # of users, individual share only decreases

Sp

open

≤j

≤2j

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A set of clients D.A source node s.cij is the distance between any pair of

nodes.(assume cij satisfies triangle inequality)

Single source rent or buy

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Single source rent or buy1) Pick a path from every client j to source s.Goal: Minimize the sum of edge costs: ΣeE min(pe,M) ce

#paths using edge e # paths

cost of eM

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Plan of attack• Gather clients into groups of M

(often done by a facility location algorithm)• Build a Steiner tree on the gathering

pointsJain&Vazirani gave cost sharing fn for Steiner treeHave cost sharing for facility locationWhy 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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Growing ghostsGrow 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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Cost sharesEach Steiner component C needs $1/second for growth.Maintenance cost of C split among users connectedUser connected to multiple components pays only to largest componentUser connected to root stops paying j =∫ fj(t) dt

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Easy factsFact: The cost shares j are cross-monotonic.Pf: More users causes more gathering points to open, more “area” is covered by Steiner clusters, clusters are bigger each j can only grow slower&stop sooner. Fact [competitiveness]: ΣjS j ≤ 2c*(S).Pf: LP duality.

Again, cost recovery is the hard part.

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Cost recoveryTo 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 treeWe 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 Ω(n) approx•Applications of cost sharing to design of approximation algorithms. Thank you!