Cost Sharing and Approximation

1. A-exam Cost Sharing and Approximation Martin Pál joint work withÉva Tardos Martin Pál: Cost sharing & Approx

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

3. cost Steiner tree # users cost No sharing # users cost Rent or buy M # users How much does it pay to share? Martin Pál: Cost sharing & Approx

4. 4 3 4 2 2 3 ? 3 1 0 4 4 2 4 3 ? 3 2 3 ? Cost shares Cost shares Steiner tree: how to split cost Martin Pál: Cost sharing & Approx

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

6. u1’=5 p1=15 p1=0 Problem: users may not reveal true utilities u2=25 p2=15 p2=22.5 u3=25 p3=15 p3=22.5 Utilities uj – utility of agent j u1=25 $45 internet Martin Pál: Cost sharing & Approx

7. 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 jS Martin Pál: Cost sharing & Approx

8. ΣjS pj ≤ c*(S) (competitiveness) ΣjS pj ≥ c(S)/β (approx. cost recovery) Desirable properties of mech’s • ΣjS 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) Martin Pál: Cost sharing & Approx

9. Cost sharing function • ξ : 2UU ℛ • ξ(S,j) – cost share of user j, given set S • Competitiveness: ΣjSξ(S,j)≤c*(S) • Cost recovery: c(S)/β≤ΣjSξ(S,j) • Voluntary particpiation: ξ(S,j) = 0 if jS • Cross-monotonicity: for jST • ξ(S,j) ≥ ξ(T,j) Martin Pál: Cost sharing & Approx

10. The Moulin&Shenker mechanism Let ξ : 2U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 pj = ξ(S,j) Thm: [Moulin&Shenker]: ξ(.) cross-monotonic  mech. group strategyproof Martin Pál: Cost sharing & Approx

11. 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 Martin Pál: Cost sharing & Approx

12. Facility Location F 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 Martin Pál: Cost sharing & Approx

13. 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: ΣiF’fi + ΣjS c(j,σ(j)) Martin Pál: Cost sharing & Approx

14. =4 =4 =6 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) Martin Pál: Cost sharing & Approx

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

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

17. 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]: ΣjSj≤c*(S). Pf: j is a feasible LP dual. Hard part: cost recovery. Martin Pál: Cost sharing & Approx

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

19. Well funded p q ≤tp1 ≤tp ≤tp1 ... tp tp1 ≤ j tq j≤tp/3 healthy  c(p,q’) ≤tp q’ 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 Martin Pál: Cost sharing & Approx

20. Fact 2: p open  clients in Sp can pay 1/3 their connection + facility cost Pf: fj = ΣjS(p)tp – cjp and j≥tp/3 Sp Fact 3: j is in no Sp can pay for 1/3 of connection Pf: fj = ΣjS(p)tp – cjp and j≥tp/3 ≤j ≤2j open Cost recovery Martin Pál: Cost sharing & Approx

21. Sp ≤j ≤2j open 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 Martin Pál: Cost sharing & Approx

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

23. #paths using edge e cost of e M # paths Single source rent or buy 1) Pick a path from every client j to source s. Goal: Minimize the sum of edge costs: ΣeE min(pe,M) ce Martin Pál: Cost sharing & Approx

24. 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? Martin Pál: Cost sharing & Approx

25. “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 Martin Pál: Cost sharing & Approx

26. 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 Martin Pál: Cost sharing & Approx

27. 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 Martin Pál: Cost sharing & Approx

28. 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]: ΣjSj≤2c*(S). Pf: LP duality. Again, cost recovery is the hard part. Martin Pál: Cost sharing & Approx

29. 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. Martin Pál: Cost sharing & Approx

30. 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. Martin Pál: Cost sharing & Approx

31. Thank you! 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. Martin Pál: Cost sharing & Approx