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An Efficient Cost Sharing Mechanism for the Prize-Collecting Steiner Forest Problem Guido Schfer Institute of Mathematics, TU Berlin, Germany Workshop on Flexible Network Design Bertinoro, October 26, 2006 joint work with: A. Gupta, J.

  1. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Alternative Efficiency Measure: Social Cost Social cost: for a set S ⊆ U , define Π ( S ) : = ∑ u i + C ( S ) i / ∈ S = ∑ u i − ∑ u i + C ( S ) = − W ( S )+ ∑ u i i ∈ U i ∈ S i ∈ U Thus: S maximizes W ( S ) iff S minimizes Π ( S ) α -approximate: approximate minimimum social cost Π ( S M ) ≤ α · Π ( S ) ∀ S ⊆ U , α ≥ 1 [Roughgarden and Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 11

  2. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Alternative Efficiency Measure: Social Cost Social cost: for a set S ⊆ U , define Π ( S ) : = ∑ u i + C ( S ) i / ∈ S = ∑ u i − ∑ u i + C ( S ) = − W ( S )+ ∑ u i i ∈ U i ∈ S i ∈ U Thus: S maximizes W ( S ) iff S minimizes Π ( S ) α -approximate: approximate minimimum social cost Π ( S M ) ≤ α · Π ( S ) ∀ S ⊆ U , α ≥ 1 [Roughgarden and Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 11

  3. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Alternative Efficiency Measure: Social Cost Social cost: for a set S ⊆ U , define Π ( S ) : = ∑ u i + C ( S ) i / ∈ S = ∑ u i − ∑ u i + C ( S ) = − W ( S )+ ∑ u i i ∈ U i ∈ S i ∈ U Thus: S maximizes W ( S ) iff S minimizes Π ( S ) α -approximate: approximate minimimum social cost Π ( S M ) ≤ α · Π ( S ) ∀ S ⊆ U , α ≥ 1 [Roughgarden and Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 11

  4. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Alternative Efficiency Measure: Social Cost Social cost: for a set S ⊆ U , define Π ( S ) : = ∑ u i + C ( S ) i / ∈ S = ∑ u i − ∑ u i + C ( S ) = − W ( S )+ ∑ u i i ∈ U i ∈ S i ∈ U Thus: S maximizes W ( S ) iff S minimizes Π ( S ) α -approximate: approximate minimimum social cost Π ( S M ) ≤ α · Π ( S ) ∀ S ⊆ U , α ≥ 1 [Roughgarden and Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 11

  5. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Cost Sharing Mechanisms β α Authors Problem Θ ( log n ) [Roughgarden, Sundararajan ’06] submodular cost 1 Θ ( log 2 n ) Steiner tree 2 Θ ( log 2 n ) [Chawla, Roughgarden, Sundarara- Steiner forest 2 jan ’06] Θ ( log n ) [Roughgarden, Sundararajan ?] facility location 3 Θ ( log 2 n ) SRoB 4 Θ ( log 2 n ) [Gupta et al. ’07] prize-collecting 3 Steiner forest Guido Schäfer Cost Sharing Mechanism for PCSF 12

  6. Cost Sharing Mechanisms State of Affairs Tricks of the Trade How to achieve β -budget balance? � � C ( S ) ≤ ∑ p i ≤ β · C ( S ) i ∈ S M Guido Schäfer Cost Sharing Mechanism for PCSF 13

  7. Cost Sharing Mechanisms State of Affairs Tricks of the Trade ... use techniques from approximation algorithms Guido Schäfer Cost Sharing Mechanism for PCSF 13

  8. Cost Sharing Mechanisms State of Affairs Tricks of the Trade How to achieve group-strategyproofness? (not everybody in the coalition is better off by misreporting his utility) Guido Schäfer Cost Sharing Mechanism for PCSF 14

  9. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Cross-Monotonic Cost Sharing Method Cost sharing method: function ξ : U × 2 U → R + ξ ( i , S ) = cost share of user i with respect to set S ⊆ U β -budget balance: C ( S ) ≤ ∑ ξ ( i , S ) ≤ β · C ( S ) ∀ S ⊆ U i ∈ S Cross-monotonicity: cost share of user i does not increase as additional users join the game: ∀ S ′ ⊆ S , ∀ i ∈ S ′ : ξ ( i , S ′ ) ≥ ξ ( i , S ) Guido Schäfer Cost Sharing Mechanism for PCSF 15

  10. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Cross-Monotonic Cost Sharing Method Cost sharing method: function ξ : U × 2 U → R + ξ ( i , S ) = cost share of user i with respect to set S ⊆ U β -budget balance: C ( S ) ≤ ∑ ξ ( i , S ) ≤ β · C ( S ) ∀ S ⊆ U i ∈ S Cross-monotonicity: cost share of user i does not increase as additional users join the game: ∀ S ′ ⊆ S , ∀ i ∈ S ′ : ξ ( i , S ′ ) ≥ ξ ( i , S ) Guido Schäfer Cost Sharing Mechanism for PCSF 15

  11. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Moulin Mechanism Given: cross-monotonic and β -budget balanced cost sharing method ξ Moulin mechanism M ( ξ ) : 1: Initialize: S M ← U 2: If for each user i ∈ S M : ξ ( i , S M ) ≤ b i then STOP 3: Otherwise, remove from S M all users with ξ ( i , S M ) > b i and repeat Thm: Moulin mechanism M ( ξ ) is a group-strategyproof cost sharing mechanism that is β -budget balanced [Moulin, Shenker ’01], [Jain, Vazirani ’01] Guido Schäfer Cost Sharing Mechanism for PCSF 16

  12. Cost Sharing Mechanisms State of Affairs Tricks of the Trade How to achieve α -approximability? � � Π ( S M ) ≤ 1 α · Π ( S ) ∀ S ⊆ U Guido Schäfer Cost Sharing Mechanism for PCSF 17

  13. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability of Cost Sharing Methods Given: arbitrary order σ on users in U Order subset S ⊆ U according to σ : S : = { i 1 ,..., i | S | } Let S j : = first j users of S α -summability: ξ is α -summable if | S | ∑ ∀ σ , ∀ S ⊆ U : ξ ( i j , S j ) ≤ α · C ( S ) j = 1 Guido Schäfer Cost Sharing Mechanism for PCSF 18

  14. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability of Cost Sharing Methods Given: arbitrary order σ on users in U Order subset S ⊆ U according to σ : S : = { i 1 ,..., i | S | } Let S j : = first j users of S α -summability: ξ is α -summable if | S | ∑ ∀ σ , ∀ S ⊆ U : ξ ( i j , S j ) ≤ α · C ( S ) j = 1 Guido Schäfer Cost Sharing Mechanism for PCSF 18

  15. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability of Cost Sharing Methods Given: arbitrary order σ on users in U Order subset S ⊆ U according to σ : S : = { i 1 ,..., i | S | } Let S j : = first j users of S α -summability: ξ is α -summable if | S | ∑ ∀ σ , ∀ S ⊆ U : ξ ( i j , S j ) ≤ α · C ( S ) j = 1 Guido Schäfer Cost Sharing Mechanism for PCSF 18

  16. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability of Cost Sharing Methods Given: arbitrary order σ on users in U Order subset S ⊆ U according to σ : S : = { i 1 ,..., i | S | } Let S j : = first j users of S α -summability: ξ is α -summable if | S | ∑ ∀ σ , ∀ S ⊆ U : ξ ( i j , S j ) ≤ α · C ( S ) j = 1 Guido Schäfer Cost Sharing Mechanism for PCSF 18

  17. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability implies Approximability Given: cross-monotonic cost sharing method ξ that satisfies β -budget balance and α -summability Thm: Moulin mechanism M ( ξ ) is a group-strategyproof cost sharing mechanism that is β -budget balanced and ( α + β ) -approximate [Roughgarden, Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 19

  18. Cost Sharing Mechanisms State of Affairs Tricks of the Trade Summability implies Approximability Given: cross-monotonic cost sharing method ξ that satisfies β -budget balance and α -summability Thm: Moulin mechanism M ( ξ ) is a group-strategyproof cost sharing mechanism that is β -budget balanced and ( α + β ) -approximate [Roughgarden, Sundararajan ’06] Guido Schäfer Cost Sharing Mechanism for PCSF 19

  19. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Part II Prize-Collecting Steiner Forest Problem Guido Schäfer Cost Sharing Mechanism for PCSF 20

  20. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Prize-Collecting Steiner Forest Problem (PCSF) Given: ◮ network N = ( V , E , c ) with edge costs c : E → R + ◮ set of n terminal pairs R = { ( s 1 , t 1 ) ,..., ( s n , t n ) } ⊆ V × V ◮ penalty π i ≥ 0 for every pair ( s i , t i ) ∈ R . Feasible solution: forest F and subset Q ⊆ R such that for all ( s i , t i ) ∈ R : either s i , t i are connected in F , or ( s i , t i ) ∈ Q Objective: compute feasible solution ( F , Q ) such that c ( F )+ π ( Q ) is minimized Guido Schäfer Cost Sharing Mechanism for PCSF 21

  21. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Prize-Collecting Steiner Forest Problem (PCSF) Given: ◮ network N = ( V , E , c ) with edge costs c : E → R + ◮ set of n terminal pairs R = { ( s 1 , t 1 ) ,..., ( s n , t n ) } ⊆ V × V ◮ penalty π i ≥ 0 for every pair ( s i , t i ) ∈ R . Feasible solution: forest F and subset Q ⊆ R such that for all ( s i , t i ) ∈ R : either s i , t i are connected in F , or ( s i , t i ) ∈ Q Objective: compute feasible solution ( F , Q ) such that c ( F )+ π ( Q ) is minimized Guido Schäfer Cost Sharing Mechanism for PCSF 21

  22. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Prize-Collecting Steiner Forest Problem (PCSF) Given: ◮ network N = ( V , E , c ) with edge costs c : E → R + ◮ set of n terminal pairs R = { ( s 1 , t 1 ) ,..., ( s n , t n ) } ⊆ V × V ◮ penalty π i ≥ 0 for every pair ( s i , t i ) ∈ R . Feasible solution: forest F and subset Q ⊆ R such that for all ( s i , t i ) ∈ R : either s i , t i are connected in F , or ( s i , t i ) ∈ Q Objective: compute feasible solution ( F , Q ) such that c ( F )+ π ( Q ) is minimized Guido Schäfer Cost Sharing Mechanism for PCSF 21

  23. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique PCSF in a Cost Sharing Context ◮ every user is associated with a terminal pair: U = R ◮ user i wants to connect s i and t i ◮ service provider can either build this connection himself, or buy connection at a price of π i from another provider ◮ cost function C ( S ) for user set S ⊆ U is given by the cost of an optimal solution for PCSF(S) Guido Schäfer Cost Sharing Mechanism for PCSF 22

  24. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique PCSF in a Cost Sharing Context ◮ every user is associated with a terminal pair: U = R ◮ user i wants to connect s i and t i ◮ service provider can either build this connection himself, or buy connection at a price of π i from another provider ◮ cost function C ( S ) for user set S ⊆ U is given by the cost of an optimal solution for PCSF(S) Guido Schäfer Cost Sharing Mechanism for PCSF 22

  25. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique PCSF in a Cost Sharing Context ◮ every user is associated with a terminal pair: U = R ◮ user i wants to connect s i and t i ◮ service provider can either build this connection himself, or buy connection at a price of π i from another provider ◮ cost function C ( S ) for user set S ⊆ U is given by the cost of an optimal solution for PCSF(S) Guido Schäfer Cost Sharing Mechanism for PCSF 22

  26. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique PCSF in a Cost Sharing Context ◮ every user is associated with a terminal pair: U = R ◮ user i wants to connect s i and t i ◮ service provider can either build this connection himself, or buy connection at a price of π i from another provider ◮ cost function C ( S ) for user set S ⊆ U is given by the cost of an optimal solution for PCSF(S) Guido Schäfer Cost Sharing Mechanism for PCSF 22

  27. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Our Results ◮ cost sharing method ξ that is cross-monotonic and 3-budget balanced for PCSF Byproduct: simple primal-dual 3-approximate algorithm ◮ reduction technique that shows that Moulin mechanism M ( ξ ) is Θ ( log 2 n ) -approximate ◮ simple proof of O ( log 3 n ) -summability for Steiner forest cost sharing method joint work with: A. Gupta, J. Könemann, S. Leonardi, R. Ravi to appear in SODA 2007 Guido Schäfer Cost Sharing Mechanism for PCSF 23

  28. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique LP Formulation c e · x e + ∑ ∑ π ( u , ¯ u ) · x u ¯ min u e ∈ E ( u , ¯ u ) ∈ R ∑ x e + x u ¯ ∀ S ∈ S , ∀ ( u , ¯ u ) ⊙ S s.t. u ≥ 1 e ∈ δ ( S ) x e ≥ 0 ∀ e ∈ E x u ¯ ∀ ( u , ¯ u ) ∈ R u ≥ 0 S = set of all Steiner cuts (separate at least one pair) δ ( S ) = edges that cross cut defined by S ( u , ¯ u ) ⊙ S = terminal pair ( u , ¯ u ) separated by S Guido Schäfer Cost Sharing Mechanism for PCSF 24

  29. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Dual LP S ∈ S ∑ ∑ ξ S , u ¯ max u ( u , ¯ u ) ⊙ S S : e ∈ δ ( S ) ∑ ∑ u ≤ c e ∀ e ∈ E ξ S , u ¯ s.t. ( u , ¯ u ) ⊙ S ∑ u ≤ π ( u , ¯ u ) ∀ ( u , ¯ u ) ∈ R ξ S , u ¯ S : ( u , ¯ u ) ⊙ S ∀ S ∈ S , ∀ ( u , ¯ u ) ⊙ S ξ S , u ¯ u ≥ 0 u = cost share that ( u , ¯ u ) receives from Steiner cut S ξ S , u ¯ Guido Schäfer Cost Sharing Mechanism for PCSF 25

  30. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Dual LP — Simplified ∑ ξ u ¯ ξ S , u ¯ (total cost share of ( u , ¯ u ) ) u : = u S : ( u , ¯ u ) ⊙ S y S : = ∑ (total dual of Steiner cut S ) ξ S , u ¯ u ( u , ¯ u ) ⊙ S ∑ y S max S ∈ S ∑ y S ≤ c e ∀ e ∈ E s.t. S : e ∈ δ ( S ) u ≤ π ( u , ¯ u ) ∀ ( u , ¯ u ) ∈ R ξ u ¯ ξ S , u ¯ ∀ S ∈ S , ∀ ( u , ¯ u ) ⊙ S u ≥ 0 Guido Schäfer Cost Sharing Mechanism for PCSF 26

  31. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Visualizing the Dual ◮ dual y S of Steiner cut S is visualized as moat around S of radius y S y S S ◮ edge e is tight if 1 ∑ y S = c e e S : e ∈ δ ( S ) 1 ◮ growth of moat corresponds to an increase in the dual value Guido Schäfer Cost Sharing Mechanism for PCSF 27

  32. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Visualizing the Dual ◮ dual y S of Steiner cut S is visualized as moat around S of radius y S y S S ◮ edge e is tight if 1 ∑ y S = c e e S : e ∈ δ ( S ) 1 S ¯ y ¯ ◮ growth of moat corresponds to an S increase in the dual value Guido Schäfer Cost Sharing Mechanism for PCSF 27

  33. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Visualizing the Dual ◮ dual y S of Steiner cut S is visualized as moat around S of radius y S y S S ◮ edge e is tight if 1 ∑ y S = c e e S : e ∈ δ ( S ) 1 S ¯ y ¯ ◮ growth of moat corresponds to an S increase in the dual value Guido Schäfer Cost Sharing Mechanism for PCSF 27

  34. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Activity Notion Death time: let d G ( u , ¯ u ) be distance between u , ¯ u in G u ) : = 1 d ( u , ¯ 2 d G ( u , ¯ u ) Activity: terminal u ∈ R is active at time τ iff ξ τ u < π ( u , ¯ u ) τ ≤ d ( u , ¯ u ) . and u ¯ Call a moat active if it contains at least one active terminal Guido Schäfer Cost Sharing Mechanism for PCSF 28

  35. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Activity Notion Death time: let d G ( u , ¯ u ) be distance between u , ¯ u in G u ) : = 1 d ( u , ¯ 2 d G ( u , ¯ u ) Activity: terminal u ∈ R is active at time τ iff ξ τ u < π ( u , ¯ u ) τ ≤ d ( u , ¯ u ) . and u ¯ Call a moat active if it contains at least one active terminal Guido Schäfer Cost Sharing Mechanism for PCSF 28

  36. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Activity Notion Death time: let d G ( u , ¯ u ) be distance between u , ¯ u in G u ) : = 1 d ( u , ¯ 2 d G ( u , ¯ u ) Activity: terminal u ∈ R is active at time τ iff ξ τ u < π ( u , ¯ u ) τ ≤ d ( u , ¯ u ) . and u ¯ Call a moat active if it contains at least one active terminal Guido Schäfer Cost Sharing Mechanism for PCSF 28

  37. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  38. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  39. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  40. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  41. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  42. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Algorithm ◮ process over time ◮ at every time τ : grow all active moats uniformly ◮ share dual growth of a moat evenly among active terminals contained in it ◮ if two active moats collide: add all new tight edges on path between them to the forest F ◮ if a terminal pair ( u , ¯ u ) becomes inactive since its cost share reaches its penalty, add ( u , ¯ u ) to the set Q ◮ terminate if all moats are inactive Guido Schäfer Cost Sharing Mechanism for PCSF 29

  43. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 0 . 5 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 1 1 1 1 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  44. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 1 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 2 2 2 2 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  45. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 1 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 2 2 2 2 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  46. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 2 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 4 2 4 2 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  47. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 2 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 4 2 4 2 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  48. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 3 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 5 2 6 2 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  49. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 4 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 5 1 8 1 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  50. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Illustration s 4 s 3 t 4 τ = 4 t 3 t 1 s 1 s 2 t 2 ( s 1 , t 1 ) ( s 2 , t 2 ) ( s 3 , t 3 ) ( s 4 , t 4 ) d ( · ) 4 1 22 3 π ( · ) ∞ 5 5 2 ξ τ 5 1 8 1 Guido Schäfer Cost Sharing Mechanism for PCSF 30

  51. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Two Quick Proofs Lem: ξ is cross-monotonic Proof (idea): at every time τ and for any S ⊆ S ′ ◮ moat system wrt. S is a refinement of moat system wrt. S ′ ◮ cost share of u wrt. S is at least cost share of u wrt. S ′ Lem: ξ is 3-budget balanced Proof (idea): ◮ cost of solution is at most 2 ∑ y S for Steiner forest and ∑ ξ u ¯ u for total penalty ◮ need to prove that ∑ y S ≤ C ( S ) (hard part) Guido Schäfer Cost Sharing Mechanism for PCSF 31

  52. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Two Quick Proofs Lem: ξ is cross-monotonic Proof (idea): at every time τ and for any S ⊆ S ′ ◮ moat system wrt. S is a refinement of moat system wrt. S ′ ◮ cost share of u wrt. S is at least cost share of u wrt. S ′ Lem: ξ is 3-budget balanced Proof (idea): ◮ cost of solution is at most 2 ∑ y S for Steiner forest and ∑ ξ u ¯ u for total penalty ◮ need to prove that ∑ y S ≤ C ( S ) (hard part) Guido Schäfer Cost Sharing Mechanism for PCSF 31

  53. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Two Quick Proofs Lem: ξ is cross-monotonic Proof (idea): at every time τ and for any S ⊆ S ′ ◮ moat system wrt. S is a refinement of moat system wrt. S ′ ◮ cost share of u wrt. S is at least cost share of u wrt. S ′ Lem: ξ is 3-budget balanced Proof (idea): ◮ cost of solution is at most 2 ∑ y S for Steiner forest and ∑ ξ u ¯ u for total penalty ◮ need to prove that ∑ y S ≤ C ( S ) (hard part) Guido Schäfer Cost Sharing Mechanism for PCSF 31

  54. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Two Quick Proofs Lem: ξ is cross-monotonic Proof (idea): at every time τ and for any S ⊆ S ′ ◮ moat system wrt. S is a refinement of moat system wrt. S ′ ◮ cost share of u wrt. S is at least cost share of u wrt. S ′ Lem: ξ is 3-budget balanced Proof (idea): ◮ cost of solution is at most 2 ∑ y S for Steiner forest and ∑ ξ u ¯ u for total penalty ◮ need to prove that ∑ y S ≤ C ( S ) (hard part) Guido Schäfer Cost Sharing Mechanism for PCSF 31

  55. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Partitioning Lemma Given: cross-monotonic cost sharing method ξ on U that is β -budget balanced for C · Lem: If there is a partition U = U 1 ∪ U 2 such that the Moulin mechanism M ( ξ ) is α i -approximate on U i for all i ∈ { 1 , 2 } , then M ( ξ ) is ( α 1 + α 2 ) β -approximate on U Guido Schäfer Cost Sharing Mechanism for PCSF 32

  56. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Partitioning Lemma Given: cross-monotonic cost sharing method ξ on U that is β -budget balanced for C · Lem: If there is a partition U = U 1 ∪ U 2 such that the Moulin mechanism M ( ξ ) is α i -approximate on U i for all i ∈ { 1 , 2 } , then M ( ξ ) is ( α 1 + α 2 ) β -approximate on U Guido Schäfer Cost Sharing Mechanism for PCSF 32

  57. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique High-Utility Users U 1 = set of all users i with u i ≥ π i Lem: (High-Utility Lemma): M ( ξ ) is 1-approximate on U 1 . Proof: By construction, ξ ( i , S ) ≤ π i ≤ u i for all i , for all S ⊆ U 1 . Thus, set S M output by Moulin mechanism M ( ξ ) is U . Moreover, U minimizes social cost. Guido Schäfer Cost Sharing Mechanism for PCSF 33

  58. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique High-Utility Users U 1 = set of all users i with u i ≥ π i Lem: (High-Utility Lemma): M ( ξ ) is 1-approximate on U 1 . Proof: By construction, ξ ( i , S ) ≤ π i ≤ u i for all i , for all S ⊆ U 1 . Thus, set S M output by Moulin mechanism M ( ξ ) is U . Moreover, U minimizes social cost. Guido Schäfer Cost Sharing Mechanism for PCSF 33

  59. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique High-Utility Users U 1 = set of all users i with u i ≥ π i Lem: (High-Utility Lemma): M ( ξ ) is 1-approximate on U 1 . Proof: By construction, ξ ( i , S ) ≤ π i ≤ u i for all i , for all S ⊆ U 1 . Thus, set S M output by Moulin mechanism M ( ξ ) is U . Moreover, U minimizes social cost. Guido Schäfer Cost Sharing Mechanism for PCSF 33

  60. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique High-Utility Users U 1 = set of all users i with u i ≥ π i Lem: (High-Utility Lemma): M ( ξ ) is 1-approximate on U 1 . Proof: By construction, ξ ( i , S ) ≤ π i ≤ u i for all i , for all S ⊆ U 1 . Thus, set S M output by Moulin mechanism M ( ξ ) is U . Moreover, U minimizes social cost. Guido Schäfer Cost Sharing Mechanism for PCSF 33

  61. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique High-Utility Users U 1 = set of all users i with u i ≥ π i Lem: (High-Utility Lemma): M ( ξ ) is 1-approximate on U 1 . Proof: By construction, ξ ( i , S ) ≤ π i ≤ u i for all i , for all S ⊆ U 1 . Thus, set S M output by Moulin mechanism M ( ξ ) is U . Moreover, U minimizes social cost. Guido Schäfer Cost Sharing Mechanism for PCSF 33

  62. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users U 2 = set of all users i with u i < π i ξ ′ = cross-monotonic cost sharing method for Steiner forest problem Similarity Property: For every S ⊆ U 2 : If there is a user i ∈ S with ξ ( i , S ) > u i or ξ ′ ( i , S ) > u i then there exists a user j ∈ S with ξ ( j , S ) > u j and ξ ′ ( j , S ) > u j . Lem: When starting with a low-utility set S ⊆ U 2 , the final user sets produced by M ( ξ ) and M ( ξ ′ ) are the same Guido Schäfer Cost Sharing Mechanism for PCSF 34

  63. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users U 2 = set of all users i with u i < π i ξ ′ = cross-monotonic cost sharing method for Steiner forest problem Similarity Property: For every S ⊆ U 2 : If there is a user i ∈ S with ξ ( i , S ) > u i or ξ ′ ( i , S ) > u i then there exists a user j ∈ S with ξ ( j , S ) > u j and ξ ′ ( j , S ) > u j . Lem: When starting with a low-utility set S ⊆ U 2 , the final user sets produced by M ( ξ ) and M ( ξ ′ ) are the same Guido Schäfer Cost Sharing Mechanism for PCSF 34

  64. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users U 2 = set of all users i with u i < π i ξ ′ = cross-monotonic cost sharing method for Steiner forest problem Similarity Property: For every S ⊆ U 2 : If there is a user i ∈ S with ξ ( i , S ) > u i or ξ ′ ( i , S ) > u i then there exists a user j ∈ S with ξ ( j , S ) > u j and ξ ′ ( j , S ) > u j . Lem: When starting with a low-utility set S ⊆ U 2 , the final user sets produced by M ( ξ ) and M ( ξ ′ ) are the same Guido Schäfer Cost Sharing Mechanism for PCSF 34

  65. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users U 2 = set of all users i with u i < π i ξ ′ = cross-monotonic cost sharing method for Steiner forest problem Similarity Property: For every S ⊆ U 2 : If there is a user i ∈ S with ξ ( i , S ) > u i or ξ ′ ( i , S ) > u i then there exists a user j ∈ S with ξ ( j , S ) > u j and ξ ′ ( j , S ) > u j . Lem: When starting with a low-utility set S ⊆ U 2 , the final user sets produced by M ( ξ ) and M ( ξ ′ ) are the same Guido Schäfer Cost Sharing Mechanism for PCSF 34

  66. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  67. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  68. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  69. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  70. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  71. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  72. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  73. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  74. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  75. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Low-Utility Users Lem: (Low-Utility Lemma): M ( ξ ) is α -approximate on U 2 if M ( ξ ′ ) is α -approximate on U 2 Proof: Solution for set with minimum social cost never pays a penalty, as u i < π i . Thus, optimal social cost for PCSF and SF are the same. Furthermore, C ( S ) ≤ C ′ ( S ) for all S ⊆ U 2 . Due to the similarity property, both mechanisms output the same set S . ′ ∗ = α Π ∗ Π ( S ) = u ( U \ S )+ C ( S ) ≤ u ( U \ S )+ C ′ ( S ) = Π ′ ( S ) ≤ α Π Guido Schäfer Cost Sharing Mechanism for PCSF 35

  76. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Putting the Pieces together... We showed: ◮ M ( ξ ) is 1-approximate on high-utility users ◮ M ( ξ ) is Θ ( log 2 n ) -approximate on low-utility users Thm: M ( ξ ) is a group-strategyproof cost sharing mechanism for PCSF that is 3-budget balanced and Θ ( log 2 n ) -approximate Remark: technique extends to other prize-collecting problems, e.g., prize-collecting facility location Guido Schäfer Cost Sharing Mechanism for PCSF 36

  77. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Putting the Pieces together... We showed: ◮ M ( ξ ) is 1-approximate on high-utility users ◮ M ( ξ ) is Θ ( log 2 n ) -approximate on low-utility users Thm: M ( ξ ) is a group-strategyproof cost sharing mechanism for PCSF that is 3-budget balanced and Θ ( log 2 n ) -approximate Remark: technique extends to other prize-collecting problems, e.g., prize-collecting facility location Guido Schäfer Cost Sharing Mechanism for PCSF 36

  78. Prize-Collecting Steiner Forest Cost Sharing Method Reduction Technique Putting the Pieces together... We showed: ◮ M ( ξ ) is 1-approximate on high-utility users ◮ M ( ξ ) is Θ ( log 2 n ) -approximate on low-utility users Thm: M ( ξ ) is a group-strategyproof cost sharing mechanism for PCSF that is 3-budget balanced and Θ ( log 2 n ) -approximate Remark: technique extends to other prize-collecting problems, e.g., prize-collecting facility location Guido Schäfer Cost Sharing Mechanism for PCSF 36

  79. Conclusions and Open Problems Part III Conclusions and Open Problems Guido Schäfer Cost Sharing Mechanism for PCSF 37

  80. Conclusions and Open Problems Conclusions New efficiency measure: ◮ circumvents classical intractability results ◮ enables to differentiate the solution quality of different cost sharing mechanisms ◮ motivates the design of “good” cost sharing mechanisms ◮ ... but still might be too restrictive!? Obs: Suppose that there is a set S ⊆ U with C ( S ′ ) ≥ C ( S ) / δ for all S ′ ⊆ S and some constant δ ≥ 1. Then there is no Ω ( log | S | ) -approximate Moulin mechanism that satisfies cost recovery. [Brenner, S. 06] Guido Schäfer Cost Sharing Mechanism for PCSF 38

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