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)

Sx

ts

c(e)xEe

e

..

min

LxelEe

e

)(

52

3

10

2

1

1

4

37

71

,5

,1

,1

, 1

,3

,3

,7

,5

,1

,1,3,5

L = 15

NP-hard

Multi-Objective Optimization & Mechanism Design

Design an efficient truthful mechanism

Utilitarian problem! ... but cannot use

VCG mechanism Sufficient property:

monotone algorithm [LOS02, BKV05]Sx

ts

c(e)xEe

e

..

min

LxelEe

e

)(

Unknown

10, 111, 10

Unknown

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

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

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 ƐL

Not monotone

A closer look at Lagrangian relaxation

Sxts

Lxelc(e)xEe

eEe

e

..

))((min

Sx

ts

c(e)xEe

e

..

min

LxelEe

e

)(

λ-OPT ≤ OPT (For feasible BMSTs and λ≥0)

Optimal Lagrangian multiplier:

OPTλ 0* maxarg

52

3

10

2

1

1

4

37 + 5λ

71

+5λ + λ

+ λ

+λ

+3λ

+3λ

+7λ

+ λ

+λ+3 λ +5λ

Geometric interpretation of λ-OPT

λ

λ -OPT

λ*

[RG96] output a positive-slope line adjacent to a negative-slope line

(1,2)-approximate solution

Adjacency relation of trees

))(()( Lelc(e)TTeTe

Monotone Lagrangian relaxation

λ

λ -OPT

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-slope negative-slope

))(()( Lelc(e)TTeTe

Returning negative-slope line is monotone (Idea)

λ

λ -OPT

λ*(λ’)*

(λ’)*-OPT

e

Output a negative-slope line adjacent to a positive-slope line

(OPT+cmax,1)-approximate solution

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

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!

“Bid-independent” Pruning

S subset of edges of size 1/Ɛ

g: S → cmin cmax

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.

“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: OPT1/Ɛ → cmin cmax

OPT1/Ɛ heaviest 1/Ɛ edges of OPT

(1+Ɛ,1)-approximate solution

“Bid-independent” Pruning: monotonicityUse 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

“Bid-independent” Pruning: Bitonicity

cmin cmax

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()

in out

is monotone!

Overall algorithm:

Or not?

Composing bitonic algorithms

cmin cmax

......

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

≈

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

monotone

Not efficient

“Efficient” Bitonic Lagrangian algorithmLagrangian based algorithm is bitonic if we return the maximum-cost negative-slope line in the set of optimal Lagrangian solutions.

λ

λ -OPT

λ*

Mechanism

Ar1 Ark...

Randomly perturb the input

just two lines at any point

Las Vegas Universally truthful PTAS for BMST

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