1. Utilitarian Mechanism Design for Multi-Objective Optimization Fabrizio Grandoni (U. Tor Vergata, Roma) Piotr Krysta (U. of Liverpool) Stefano Leonardi (U. La Sapienza, Roma) Carmine Ventre (U. of Liverpool)

2. Multi-Objective Optimization: Budgeted MST (BMST) 5 2 3 10 2 1 1 4 3 7 7 1,5,1,3,7,5,1,3,5 L = 15 NP-hard

3. Multi-Objective Optimization & Mechanism Design Design an efficient truthful mechanism Utilitarian problem! ... but cannot use VCG mechanism Sufficient property: monotone algorithm [LOS02, BKV05] Unknown 10, 111, 10 Unknown

4. Monotone Algorithms c(e) Algorithm A is monotone if for each agent (edge) e, fixed bids of all agents but e, we have: A selects e l(e) e is selected by A Design a monotone algorithm for BMST

5. Monotone algorithms for BMST FPTAS that return solutions violating the budget by at most a factor of (1 + Ɛ )  Making the computation of approximate Pareto curves by [Papadimitriou&Yannakakis, 00] monotone Randomized PTAS that return feasible solutions  Making Lagrangian-relaxation technique monotone

6. PTAS for BMST [RG96] Idea 1: Solve Lagrangian Relaxation of BMST  Obtain a (1,2)-approximate solution Solution of optimal cost but of length at most 2L Idea 2: Guess the 1/ Ɛ longest edges of OPT, prune edges with length higher than Ɛ L Not monotone

7. A closer look at Lagrangian relaxation λ-OPT ≤ OPT (For feasible BMSTs and λ≥0) Optimal Lagrangian multiplier: 5 2 3 10 2 1 1 4 3 7 + 5λ 7 1 +5λ + λ+ λ + λ+ λ +λ+λ +3λ +7λ + λ+ λ +λ+λ +3 λ +5λ

8. Geometric interpretation of λ-OPT λ λ -OPT λ*λ* [RG96] output a positive-slope line adjacent to a negative-slope line (1,2)-approximate solution Adjacency relation of trees

9. Monotone Lagrangian relaxation λ λ -OPT λ*λ* e e e l’(e) < l(e) (λ’)* By lowering l value e is not selected anymore: [RG96] is not monotone Output a line adjacent to a linepositive-slopenegative-slope

10. Returning negative-slope line is monotone (Idea) λ λ -OPT λ*λ* (λ’)* (λ’)*-OPT e Output a negative-slope line adjacent to a positive-slope line (OPT+cmax,1)-approximate solution

11. Monotone(?) PTAS for BMST (inspired by [RG96]) Idea 1: Solve Lagrangian Relaxation of BMST  Obtain a (OPT+cmax,1)-approximate solution Idea 2: Guess the 1/ Ɛ heaviest edges of OPT, prune edges with cost higher than the minimum cost in the guess monotone Not monotone

12. Guessing is inherently not monotone...... if a selected edge lowers her cost too much...... we prune all the edges from the graph and no solution is output! Pruning must be (somehow) independent from the actual declaration!

13. “Bid-independent” Pruning S subset of edges of size 1/ Ɛ g: S → cmincmax powers of 1+ Ɛ Use any such g (i.e., any S and any assignment of powers of 1+ Ɛ as costs to elements of S) as a guess, run Lagrangian- based algorithm and take the minimum-cost solution among those.

14. “Bid-independent” Pruning: approximation guarantee Use any such g (i.e., any S and any assignment of powers of 1+ Ɛ as costs to elements of S) as a guess, run Lagrangian-based algorithm and take the minimum-cost solution among those. g: OPT 1/ Ɛ → cmincmax OPT 1/ Ɛ heaviest 1/ Ɛ edges of OPT (1+ Ɛ,1)-approximate solution

15. “Bid-independent” Pruning: monotonicity Use any such g (i.e., any S and any assignment of powers of 1+ Ɛ as costs to elements of S) as a guess, run Lagrangian-based algorithm and take the minimum-cost solution among those. Composition of monotone algorithms is not monotone [MN02]...... but a “fixed*” composition of bitonic algorithms is! [MN02, BKV05] * bid-independent

16. “Bid-independent” Pruning: Bitonicity cmincmax cmin ’ cmax ’ Lagrangian-based algorithm is bitonic if we return the maximum- cost negative-slope line in the set of optimal lagrangian solutions Run Lagrangian-based algorithm for all powers of (1+ Ɛ ) between cmin and cmax for any guess. bid c() inout is monotone! Overall algorithm: Or not?

17. Composing bitonic algorithms cmincmax... Actual Algorithm: Run Lagrangian-based algorithm for all powers of (1+ Ɛ ) between cmin and cmax for any guess. Ideal Algorithm: Run Lagrangian-based algorithm for all powers of (1+ Ɛ ) for any guess. Whole graphEmpty graph ≈

18. Monotone P(?)TAS for BMST (inspired by [RG96]) Idea 1: Solve Lagrangian Relaxation of BMST  Obtain a (OPT+cmax,1)-approximate solution Idea 2: Guess the 1/ Ɛ heaviest edges of OPT, prune edges with cost higher than the minimum cost in the guess monotone Not efficient

19. “Efficient” Bitonic Lagrangian algorithm Lagrangian based algorithm is bitonic if we return the maximum-cost negative-slope line in the set of optimal Lagrangian solutions. λ λ -OPT λ*λ* Mechanism A r1 A rk... Randomly perturb the input just two lines at any point Las Vegas Universally truthful PTAS for BMST

20. Conclusions Las Vegas universally truthful PTAS for BMST inspired by [RG96]  Output negative instead of positive slope lines Sensitivity analysis of LPs to show monotonicity  Novel monotone guessing step Making the Lagrangian algorithm bitonic  Truthfulness “only” in the universal sense Input perturbation (Not showed) Monotone FPTASs for certain general multi-objective optimization problems