14 Oct 03Cost sharing & Approximation1 Group strategy proof mechanisms via primal-dual algorithms (Cost Sharing) Martin PálÉva Tardos

14 Oct 03Cost sharing & Approximation2 Our setting Universe U of (selfish) users Users want to benefit from shared infrastructure cost function c*() c*(U) = cost of c*( ) = cost of c*(S) – cost of optimal infrastructure for users in S often NP-hard to compute U = {,,,, }

14 Oct 03Cost sharing & Approximation3 Sharing the cost costuser shares Σ i  U p i ≥ c*(U)Cost Recovery: Approximate  Known as core in game theory often empty for games of interest p +p +p ≤ c( ) p +p +p +p +p ≥ c( ) Σ i  S p i ≤ c * (S)Competitiveness: no overpaymentfor any S  U:

14 Oct 03Cost sharing & Approximation4 Modeling the users u i – value of service for i User may say NO if p i >u i util(i) = u i – p i if i gets service 0 otherwise need to recompute tree & payments! Cost sharing fn: ξ : 2 U  U   + ξ(S,i) – share of user i, if set S served ξ(U, ) ≤ ξ(U-, ) cross-monotonic: for i  S  T  U ξ(S,i) ≥ ξ(T,i)

14 Oct 03Cost sharing & Approximation5 The Moulin&Shenker mechanism S := U repeat ask each user i “is ξ(S,i) ≤ u i ?” drop all i  S who say NO until all i  S say YES Output: set S; prices p i = ξ(S,i) Theorem: [Moulin&Shenker]: ξ(.) cross-monotonic  mechanism group strategyproof uiui cost share

14 Oct 03Cost sharing & Approximation6 Exact cross-monotonic sharing exists if c * () submodular [Moulin&Shenker 98] Exact cost sharing for spanning tree [Kent&Skorin-Kapov 96], [Jain&Vazirani 01] Any other games for which cross-mono sharing exists? Known cross-monotonic functions implies 2-approx. cost sharing for Steiner tree

14 Oct 03Cost sharing & Approximation7 General method for generating competitive, cross- monotonic cost shares using primal-dual algorithms Used our method to construct a cost sharing fn for: Our results recovers 1/3 of cost recovers 1/15 of cost competitive, cross-monotonic constructive proof gives an approximation algorithm Metric Facility Location Single Sink Rent or Buy

14 Oct 03Cost sharing & Approximation8 Primal-dual approximation for Facility Location [Jain&Vazirani99], [Mahdian,Ye&Zhang 02], [Mettu&Plaxton 00] Approximation for Single Sink Rent or Buy [Karger&Minkoff 00], [Meyerson&Munagala 00] [Swamy&Kumar 02], [Gupta,Kumar&Roughgarden 03] Approximation algorithms

14 Oct 03Cost sharing & Approximation9 Single Sink Rent or Buy U is a set of users. r is the sink (root) node. graph G with edge lengths c e. 1) Find a path from each user to sink. 2) Rent or Buy each edge. Rent: pay c e for each path using e Buy: pay M  c e Goal: minimize rental + buying cost.

14 Oct 03Cost sharing & Approximation10 Facility Location F is a set of facilities. U is a set of users. c ij is the distance between any i and j in U  F. (assume c ij satisfies triangle inequality) f i : cost of facility i

14 Oct 03Cost sharing & Approximation11 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: Σ i  F’ f i + Σ j  S c(j,σ(j))

14 Oct 03Cost sharing & Approximation12 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  =6 cost shares

14 Oct 03Cost sharing & Approximation13...do not yield cross-monotonic shares with,  ( )=8 helped to stop earlier failed to help  =3  =8 previously,  ( )=6

14 Oct 03Cost sharing & Approximation14 Ghost shares After  i freezes, continue growing its ghost  i ghosts keep growing forever  =3  =5.5 Fact: Shares  i are cross-monotonic. Pf: more users  more ghosts  facilities open sooner   i can stop growing earlier

14 Oct 03Cost sharing & Approximation15 Constructing a solution (1) S p : set of users contributing to p at time of opening – “contributor set” t p : time of opening facility p facility p is well funded, if  j ≥t p /3 for every j  S p SpSp tptp p 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(t p - t r ) q t q ≤t p /3 r t r ≤t q /3

14 Oct 03Cost sharing & Approximation16 Constructing a solution (2) Problem: user contributing to multiple well-funded facilities Solution: close all of them but one (process by increasing t p ) SpSp tptp p SqSq tqtq q Lemma: For every well funded facility p there is a nearby open q such that dist(p,q) ≤ 2t p p r ≤≤ ≤2(t p - t r )≤2t r q well-fundedopen

14 Oct 03Cost sharing & Approximation17 Summing up Every open facility is well funded, i.e. can be paid for by S p. p r ≤≤ ≤2(t p - t r )≤2t r q well-fundedopen SpSp tptp p 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.

14 Oct 03Cost sharing & Approximation18 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 Jain&Vazirani gave cost sharing fn for Steiner tree Have cost sharing for facility location Why not combine?

14 Oct 03Cost sharing & Approximation19 “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

14 Oct 03Cost sharing & Approximation20 General method for generating competitive, cross- monotonic cost shares using primal-dual algorithms. Used our method to construct a cost sharing fn for Facility Location & Single Sink Rent or Buy. Other problems admitting cross-mono cost sharing? Steiner Forest? Covering problems? Impossibility results? Conclusions

14 Oct 03Cost sharing & Approximation21 The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. The End. *** *** The End *** ***

14 Oct 03Cost sharing & Approximation22 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 =∫ f j (t) dt

14 Oct 03Cost sharing & Approximation23 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

14 Oct 03Cost sharing & Approximation24 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 2t p. ≤tp≤tp  j ≤t p /3 t p1 ≤  j ≤ t p1 tptp p... q tqtq Well funded q’ healthy c(p,q) ≤2(t p -t q ) c(q,q’)≤2 t q  c(p,q’) ≤t p

14 Oct 03Cost sharing & Approximation25 Cost Sharing Internet: many independent agents Not hostile, but selfish Willing to cooperate, if it helps them

14 Oct 03Cost sharing & Approximation26 Steiner tree: how to split cost ? ?? Cost shares

14 Oct 03Cost sharing & Approximation27 OPT( ) = 5 p( ) = 5-3 = 2 OPT( ) = 3 Steiner tree: how to split cost No “fair” cost allocation exists!

14 Oct 03Cost sharing & Approximation28 Desirable properties of mechanisms Σ j  S p j =c * (S) (budget balance) Only people in S pay (voluntary participation) No cheating, even in groups (group strategyproofness) If u j high enough, j guaranteed to be in S (consumer sovereignity) Σ j  S p j ≤ c * (S) (competitiveness) Σ j  S p j ≥ c(S)/β (approx. cost recovery)

14 Oct 03Cost sharing & Approximation29 Cost sharing function ξ : 2 U  U  ℛ ξ(S,j) – cost share of user j, given set S Competitiveness: Σ j  S ξ(S,j) ≤ c * (S) Cost recovery: c(S)/β ≤ Σ j  S ξ(S,j) Voluntary particpiation: ξ(S,j) = 0 if j  S Cross-monotonicity: for j  S  T ξ(S,j) ≥ ξ(T,j)

14 Oct 03Cost sharing & Approximation30 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.

14 Oct 03Cost sharing & Approximation31 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.

14 Oct 03Cost sharing & Approximation32 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!