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Utilitarian Mechanism Design for Multi-Objective Optimization

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). Multi-Objective Optimization: Budgeted MST (BMST). L = 15. 3. ,7. NP-hard.

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Utilitarian Mechanism Design for Multi-Objective Optimization

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

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

  3. Multi-Objective Optimization & Mechanism Design • Design an efficient truthfulmechanism • Utilitarian problem! • ... but cannot use VCG mechanism • Sufficient property: monotone algorithm [LOS02, BKV05] Unknown 10, 1 11, 10 Unknown

  4. Monotone Algorithms Algorithm A is monotone if for each agent (edge) e, fixed bids of all agents but e, we have: l(e) A selects e e is selected by A c(e) 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 LagrangianRelaxationof BMST • Obtain a (1,2)-approximatesolution • Solutionofoptimalcostbutoflength at most 2L • Idea 2: Guess the 1/Ɛ longest edges of OPT, prune edges with length higher than ƐL Not monotone

  7. A closer look at Lagrangian relaxation 3 +7λ 1 +λ 5 +λ 10 +λ 2 +5λ 2 +3λ 1 +λ 3 +3λ λ-OPT ≤ OPT 7 +5λ (ForfeasibleBMSTs and λ≥0) 7 +λ 4 +3 λ OptimalLagrangianmultiplier: 1 +5λ

  8. Geometricinterpretationofλ-OPT λ -OPT λ* λ [RG96] output a positive-slope line adjacent to a negative-slope line Adjacency relation oftrees (1,2)-approximatesolution

  9. Monotone Lagrangianrelaxation λ -OPT e l’(e) < l(e) e e (λ’)* λ* λ Output a lineadjacentto a line positive-slope negative-slope Bylowering l value e isnotselectedanymore: [RG96] isnot monotone

  10. Returningnegative-slopelineis monotone (Idea) λ -OPT e (λ’)*-OPT (λ’)* λ* λ Output a negative-slopelineadjacentto a positive-slopeline (OPT+cmax,1)-approximatesolution

  11. Monotone(?) PTAS for BMST (inspired by [RG96]) • Idea 1: Solve LagrangianRelaxationof BMST • Obtain a (OPT+cmax,1)-approximatesolution • 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 cmin cmax g: S → powers of 1+Ɛ S subset of edges of size 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. OPT1/Ɛ heaviest 1/Ɛ edges of OPT cmin cmax g: OPT1/Ɛ → (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 ascoststo 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 c() 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 in out Overallalgorithm: cmin cmax cmin’ cmax’ is monotone! Or not?

  17. Composing bitonic algorithms ≈ 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. cmin cmax ... ... Emptygraph Wholegraph

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

  19. “Efficient” BitonicLagrangian algorithm Lagrangian based algorithm is bitonic if we return the maximum-cost negative-slope line in the set of optimal Lagrangian solutions. λ -OPT Mechanism Randomly perturb the input Ar1 ... Ark just two lines at any point λ* λ 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

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