1. 12 Sep 03Martin Pál: Cost sharing & Approx1 Cost Sharing and Approximation Martin Pál joint work with Éva Tardos A-exam

2. 12 Sep 03Martin Pál: Cost sharing & Approx2 Cost Sharing Internet: many independent agents Not hostile, but selfish Willing to cooperate, if it helps them

3. 12 Sep 03Martin Pál: Cost sharing & Approx3 How much does it pay to share? # users cost No sharingSteiner tree cost # users cost M Rent or buy

4. 12 Sep 03Martin Pál: Cost sharing & Approx4 Steiner tree: how to split cost 443 22 ? ?? 1 33 0 43 3 22 3 44 Cost shares

5. 12 Sep 03Martin Pál: Cost sharing & Approx5 OPT( ) = 5 p( ) = 5-3 = 2 OPT( ) = 3 Steiner tree: how to split cost No “fair” cost allocation exists!

6. 12 Sep 03Martin Pál: Cost sharing & Approx6 Utilities $45 u 1 =25 u 2 =25 u 3 =25 internet p 1 =15 p 2 =15 p 3 =15 u j – utility of agent j u 1 ’=5 Problem: users may not reveal true utilities p 1 =0 p 2 =22.5 p 3 =22.5

7. 12 Sep 03Martin Pál: Cost sharing & Approx7 Cost sharing mechanism..is a protocol/algorithm that requests utility u j from every user j selects the set S of people serviced builds network servicing S Computes the payment p j of every user j  S

8. 12 Sep 03Martin Pál: Cost sharing & Approx8 Desirable properties of mech’s 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)

9. 12 Sep 03Martin Pál: Cost sharing & Approx9 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)

10. 12 Sep 03Martin Pál: Cost sharing & Approx10 The Moulin&Shenker mechanism Let ξ : 2 U  U  ℛ be a cost sharing function. S  U while unhappy users exist offer each j  S service at price ξ(S,j) S  S – {users j who rejected} output the set S and prices p j = ξ(S,j) Thm: [Moulin&Shenker]: ξ(.) cross-monotonic  mech. group strategyproof

11. 12 Sep 03Martin Pál: Cost sharing & Approx11 Designing x-mono functions We construct cross-monotonic cost shares for two games: Metric facility location game Single source rent or buy game Facility location: competitive, recovers 1/3 of cost Rent or buy:competitive, recovers 1/15 of cost

12. 12 Sep 03Martin Pál: Cost sharing & Approx12 Facility Location F is a set of facilities. D is a set of clients. c ij is the distance between any i and j in D  F. (assume c ij satisfies triangle inequality) f i : cost of facility i

13. 12 Sep 03Martin Pál: Cost sharing & Approx13 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. 12 Sep 03Martin Pál: Cost sharing & Approx14 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  =6

15. 12 Sep 03Martin Pál: Cost sharing & Approx15..do not yield x-mono shares  =5 with,  ( )=6 without,  ( )=5 was stopped prematurely in the first run

16. 12 Sep 03Martin Pál: Cost sharing & Approx16 Ghost shares  =4  =5 Two shares per user: ghost share  j real share  j  j grows forever  j stops when connected

17. 12 Sep 03Martin Pál: Cost sharing & Approx17 Easy facts Fact 1: cost shares  j are cross-monotonic. Pf: More users opens facilities faster  each  j can only stop growing earlier. Fact 2 [competitiveness]: Σ j  S  j ≤ c * (S). Pf:  j is a feasible LP dual. Hard part: cost recovery.

18. 12 Sep 03Martin Pál: Cost sharing & Approx18 Constructing a solution t p : time when facility p opened S p : set of clients connected to p at time t p facility p is well funded, if 3  j ≥t p for every j  S p each facility is either poisoned or healthy p is poisoned if it shares a client with a well funded healthy facility q and t p > t q  =7  =2 t p =2 t q =7

19. 12 Sep 03Martin Pál: Cost sharing & Approx19 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

20. 12 Sep 03Martin Pál: Cost sharing & Approx20 Cost recovery Fact 2: p open  clients in S p can pay 1/3 their connection + facility cost Pf: f j = Σ j  S(p) t p – c jp and  j ≥t p /3 SpSp Fact 3: j is in no S p  can pay for 1/3 of connection Pf: f j = Σ j  S(p) t p – c jp and  j ≥t p /3 open ≤j≤j ≤2  j

21. 12 Sep 03Martin Pál: Cost sharing & Approx21 Cost recovery Summary: 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 SpSp open ≤j≤j ≤2  j

22. 12 Sep 03Martin Pál: Cost sharing & Approx22 A set of clients D. A source node s. c ij is the distance between any pair of nodes. (assume c ij satisfies triangle inequality) Single source rent or buy

23. 12 Sep 03Martin Pál: Cost sharing & Approx23 Single source rent or buy 1) Pick a path from every client j to source s. Goal: Minimize the sum of edge costs: Σ e  E min(p e,M) c e #paths using edge e # paths cost of e M

24. 12 Sep 03Martin Pál: Cost sharing & Approx24 Plan of attack Gather clients into groups of M (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?

25. 12 Sep 03Martin Pál: Cost sharing & Approx25 “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

26. 12 Sep 03Martin Pál: Cost sharing & Approx26 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

27. 12 Sep 03Martin Pál: Cost sharing & Approx27 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

28. 12 Sep 03Martin Pál: Cost sharing & Approx28 Easy facts Fact: 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]: Σ j  S  j ≤ 2c * (S). Pf: LP duality. Again, cost recovery is the hard part.

29. 12 Sep 03Martin Pál: Cost sharing & Approx29 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.

30. 12 Sep 03Martin Pál: Cost sharing & Approx30 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.

31. 12 Sep 03Martin Pál: Cost sharing & Approx31 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!