Competitive Analysis for Multi-Objective Algorithms
Competitive Analysis of Multi-Objective Online Algorithms Morten Tiedemann
Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics DFG RTG 1703
TOLA - ICALP 2014 Workshop, IT University of Copenhagen Denmark, July 7, 2014
Who gets your antique car?
2/12
Who gets your antique car?
max
b
b
b
2/12
profit appreciation
Online Optimization In online optimization, an algorithm has to make decisions based on a sequence of incoming bits of information without knowledge of future inputs.
Competitive Analysis I An algorithm alg is called c-competitive, if for all sequences σ
alg(σ) ≥
1 · opt(σ) + α. c
I The infimum over all values c such that alg is c-competitive is called
the competitive ratio of alg. I An algorithm is called competitive if it attains a “constant” competitive
ratio. 3/12
Online Optimization (contd.)
P b
b
√
end b
Pp b
b
p
4/12
b
for t = 1, . . . , T do Accept a price pt if p pt ≥ Pp
b
alg is
q
P -competitive. p
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
5/12
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
Efficient Solutions I A feasible solution xˆ ∈ X is called efficient if there is no other x ∈ X
such that f (x) f (ˆ x ), where denotes a componentwise order, i.e., for x, y ∈ Rn , x y :⇔ xi ≤ yi , for i = 1, . . . , n, and x 6= y .
5/12
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
Efficient Solutions I A feasible solution xˆ ∈ X is called efficient if there is no other x ∈ X
such that f (x) f (ˆ x ), where denotes a componentwise order, i.e., for x, y ∈ Rn , x y :⇔ xi ≤ yi , for i = 1, . . . , n, and x 6= y . I If xˆ is an efficient solution, f (ˆ x ) is called non-dominated point.
5/12
Multi-Objective Optimization (contd.)
max
b
b
b
6/12
profit appreciation
Multi-Objective Online Problem
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+ I algorithm alg I I
7/12
feasible solution alg[I ] ∈ X (I ) associated objective alg(I ) = f (I , alg[I ])
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+ I algorithm alg I I
feasible solution alg[I ] ∈ X (I ) associated objective alg(I ) = f (I , alg[I ])
I optimal algorithm opt I I
7/12
opt[I ] = {x ∈ X (I ) | x is an efficient solution to P} objective associated with x ∈ opt[I ] is denoted by opt(x)
Multi-Objective Approximation Algorithms
8/12
Multi-Objective Approximation Algorithms
ρ-approximation of a solution x fi (x 0 ) ≤ ρ · fi (x) for i = 1, . . . , n
8/12
Multi-Objective Approximation Algorithms
ρ-approximation of a solution x fi (x 0 ) ≤ ρ · fi (x) for i = 1, . . . , n
ρ-approximation of a set of efficient solutions for every feasible solution x, X 0 contains a feasible solution x 0 that is a ρ-approximation of x
8/12
Multi-Objective Online Algorithms
c-competitive A multi-objective online algorithm alg is c-competitive if for all finite input sequences I there exists an efficient solution x ∈ opt[I ] such that ALG (I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
9/12
Multi-Objective Online Algorithms
c-competitive A multi-objective online algorithm alg is c-competitive if for all finite input sequences I there exists an efficient solution x ∈ opt[I ] such that ALG (I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
strongly c-competitive A multi-objective online algorithm alg is strongly c-competitive if for all finite input sequences I and all efficient solutions x ∈ opt[I ], alg(I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
9/12
Bi-Objective Online Search
10/12
Bi-Objective Online Search I request rt = pt , qt
10/12
|
in each time period t = 1, . . . , T
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
10/12
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? or qt ≥ q ? end
q⋆
q
pt p 10/12
p⋆
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? or qt ≥ q ? end
q⋆
q
c = max pt p
10/12
p⋆
P
P Q P Q , , , p q? p? q
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? and qt ≥ q ? end
q⋆
q
pt p 10/12
p⋆
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? and qt ≥ q ? end
q⋆
q
c = max pt p
10/12
p⋆
P
P q? p? Q , , , p q p q
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt
pt · q t = z ⋆
Q
for t = 1, . . . do Accept a request rt if pt · qt ≥ z ? q⋆
end
q
pt p z⋆ Q 10/12
p⋆
z⋆ q
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt
pt · q t = z ⋆
Q
for t = 1, . . . do Accept a request rt if pt · qt ≥ z ? q⋆
end s
q
c= pt p z⋆ Q
10/12
p⋆
z⋆ q
P
PQ pq
Randomization
p2t m2 23 = M2 p1t · p2t = m1 26
···
m2 22
bc
m2 21
bc
bc
bc
p1t · p2t = m2 27
bc
bc
m1 23
m1 24
bc
m2 20 m1 20 m1 21 m1 22
11/12
m1 25 = M1
p1t
Conclusion & Further Research
max P b
b
√
b
b
b
b
Pp b
b
p
12/12
b
b
profit appreciation
Conclusion & Further Research
max P b
b
√
profit appreciation
b
b
b
b
Pp b
b
b
b
p
I application to multi-objective versions of classical online problems I relations between single- and multi-objective online optimization I alternative concepts
12/12
The Multi-Objective k-Canadian Traveller Problem
s
13/12
1 0
0 0
0 1
0 0
t
Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics DFG RTG 1703
TOLA - ICALP 2014 Workshop, IT University of Copenhagen Denmark, July 7, 2014
Who gets your antique car?
2/12
Who gets your antique car?
max
b
b
b
2/12
profit appreciation
Online Optimization In online optimization, an algorithm has to make decisions based on a sequence of incoming bits of information without knowledge of future inputs.
Competitive Analysis I An algorithm alg is called c-competitive, if for all sequences σ
alg(σ) ≥
1 · opt(σ) + α. c
I The infimum over all values c such that alg is c-competitive is called
the competitive ratio of alg. I An algorithm is called competitive if it attains a “constant” competitive
ratio. 3/12
Online Optimization (contd.)
P b
b
√
end b
Pp b
b
p
4/12
b
for t = 1, . . . , T do Accept a price pt if p pt ≥ Pp
b
alg is
q
P -competitive. p
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
5/12
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
Efficient Solutions I A feasible solution xˆ ∈ X is called efficient if there is no other x ∈ X
such that f (x) f (ˆ x ), where denotes a componentwise order, i.e., for x, y ∈ Rn , x y :⇔ xi ≤ yi , for i = 1, . . . , n, and x 6= y .
5/12
Multi-Objective Optimization
Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn , and objective vector f : X 7→ Rk : P
max f (x) s.t. x ∈ X
Efficient Solutions I A feasible solution xˆ ∈ X is called efficient if there is no other x ∈ X
such that f (x) f (ˆ x ), where denotes a componentwise order, i.e., for x, y ∈ Rn , x y :⇔ xi ≤ yi , for i = 1, . . . , n, and x 6= y . I If xˆ is an efficient solution, f (ˆ x ) is called non-dominated point.
5/12
Multi-Objective Optimization (contd.)
max
b
b
b
6/12
profit appreciation
Multi-Objective Online Problem
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+
7/12
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+ I algorithm alg I I
7/12
feasible solution alg[I ] ∈ X (I ) associated objective alg(I ) = f (I , alg[I ])
Multi-Objective Online Problem
Multi-objective (online) optimization problem P I set of inputs I I set of feasible outputs X (I ) ∈ Rn for I ∈ I I objective function f given as f : I × X 7→ Rn+ I algorithm alg I I
feasible solution alg[I ] ∈ X (I ) associated objective alg(I ) = f (I , alg[I ])
I optimal algorithm opt I I
7/12
opt[I ] = {x ∈ X (I ) | x is an efficient solution to P} objective associated with x ∈ opt[I ] is denoted by opt(x)
Multi-Objective Approximation Algorithms
8/12
Multi-Objective Approximation Algorithms
ρ-approximation of a solution x fi (x 0 ) ≤ ρ · fi (x) for i = 1, . . . , n
8/12
Multi-Objective Approximation Algorithms
ρ-approximation of a solution x fi (x 0 ) ≤ ρ · fi (x) for i = 1, . . . , n
ρ-approximation of a set of efficient solutions for every feasible solution x, X 0 contains a feasible solution x 0 that is a ρ-approximation of x
8/12
Multi-Objective Online Algorithms
c-competitive A multi-objective online algorithm alg is c-competitive if for all finite input sequences I there exists an efficient solution x ∈ opt[I ] such that ALG (I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
9/12
Multi-Objective Online Algorithms
c-competitive A multi-objective online algorithm alg is c-competitive if for all finite input sequences I there exists an efficient solution x ∈ opt[I ] such that ALG (I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
strongly c-competitive A multi-objective online algorithm alg is strongly c-competitive if for all finite input sequences I and all efficient solutions x ∈ opt[I ], alg(I ) 5 c · opt(x) + α, where α ∈ Rn is a constant vector independent of I .
9/12
Bi-Objective Online Search
10/12
Bi-Objective Online Search I request rt = pt , qt
10/12
|
in each time period t = 1, . . . , T
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
10/12
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? or qt ≥ q ? end
q⋆
q
pt p 10/12
p⋆
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? or qt ≥ q ? end
q⋆
q
c = max pt p
10/12
p⋆
P
P Q P Q , , , p q? p? q
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? and qt ≥ q ? end
q⋆
q
pt p 10/12
p⋆
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt Q
for t = 1, . . . do Accept a request rt if pt ≥ p ? and qt ≥ q ? end
q⋆
q
c = max pt p
10/12
p⋆
P
P q? p? Q , , , p q p q
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt
pt · q t = z ⋆
Q
for t = 1, . . . do Accept a request rt if pt · qt ≥ z ? q⋆
end
q
pt p z⋆ Q 10/12
p⋆
z⋆ q
P
Bi-Objective Online Search I request rt = pt , qt
|
in each time period t = 1, . . . , T
I pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q
Reservation Price Policy qt
pt · q t = z ⋆
Q
for t = 1, . . . do Accept a request rt if pt · qt ≥ z ? q⋆
end s
q
c= pt p z⋆ Q
10/12
p⋆
z⋆ q
P
PQ pq
Randomization
p2t m2 23 = M2 p1t · p2t = m1 26
···
m2 22
bc
m2 21
bc
bc
bc
p1t · p2t = m2 27
bc
bc
m1 23
m1 24
bc
m2 20 m1 20 m1 21 m1 22
11/12
m1 25 = M1
p1t
Conclusion & Further Research
max P b
b
√
b
b
b
b
Pp b
b
p
12/12
b
b
profit appreciation
Conclusion & Further Research
max P b
b
√
profit appreciation
b
b
b
b
Pp b
b
b
b
p
I application to multi-objective versions of classical online problems I relations between single- and multi-objective online optimization I alternative concepts
12/12
The Multi-Objective k-Canadian Traveller Problem
s
13/12
1 0
0 0
0 1
0 0
t
