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
7
1
,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
Sx
ts
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
7
1
+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: 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
“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
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