Competitive Analysis of Multi-Objective Online Algorithms Morten - - PowerPoint PPT Presentation

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

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Who gets your antique car?

b b b

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Who gets your antique car?

b b b

• max
• profit

appreciation

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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

◮ An algorithm alg is called c-competitive, if for all sequences σ

alg(σ) ≥ 1 c · opt(σ) + α.

◮ The infimum over all values c such that alg is c-competitive is called

the competitive ratio of alg.

◮ An algorithm is called competitive if it attains a “constant” competitive

ratio.

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Online Optimization (contd.)

b b b b b b b

√Pp P p

for t = 1, . . . , T do Accept a price pt if pt ≥

• Pp

end alg is

• P

p -competitive.

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Multi-Objective Optimization

Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn, and objective vector f : X → Rk: P max f (x) s.t. x ∈ X

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Multi-Objective Optimization

Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn, and objective vector f : X → Rk: P max f (x) s.t. x ∈ X

Efficient Solutions

◮ A feasible solution ˆ

x ∈ 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 = y.

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Multi-Objective Optimization

Consider a multi-objective optimization problem P for a given feasible set X ⊆ Rn, and objective vector f : X → Rk: P max f (x) s.t. x ∈ X

Efficient Solutions

◮ A feasible solution ˆ

x ∈ 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 = y.

◮ If ˆ

x is an efficient solution, f (ˆ x) is called non-dominated point.

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Multi-Objective Optimization (contd.)

b b b

• max
• profit

appreciation

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Multi-Objective Online Problem

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

◮ set of inputs I

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

◮ set of inputs I ◮ set of feasible outputs X(I) ∈ Rn for I ∈ I

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

◮ set of inputs I ◮ set of feasible outputs X(I) ∈ Rn for I ∈ I ◮ objective function f given as f : I × X → Rn

+

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

◮ set of inputs I ◮ set of feasible outputs X(I) ∈ Rn for I ∈ I ◮ objective function f given as f : I × X → Rn

+

◮ algorithm alg

◮ feasible solution alg[I] ∈ X(I) ◮ associated objective alg(I) = f (I, alg[I])

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Multi-Objective Online Problem

Multi-objective (online) optimization problem P

◮ set of inputs I ◮ set of feasible outputs X(I) ∈ Rn for I ∈ I ◮ objective function f given as f : I × X → Rn

+

◮ algorithm alg

◮ feasible solution alg[I] ∈ X(I) ◮ associated objective alg(I) = f (I, alg[I])

◮ optimal algorithm opt

◮ opt[I] = {x ∈ X(I) | x is an efficient solution to P} ◮ objective associated with x ∈ opt[I] is denoted by opt(x)

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Multi-Objective Approximation Algorithms

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Multi-Objective Approximation Algorithms

ρ-approximation of a solution x

fi(x′) ≤ ρ · fi(x) for i = 1, . . . , n

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Multi-Objective Approximation Algorithms

ρ-approximation of a solution x

fi(x′) ≤ ρ · fi(x) for i = 1, . . . , n

ρ-approximation of a set of efficient solutions

for every feasible solution x, X ′ contains a feasible solution x′ that is a ρ-approximation of x

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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) ≦ c · opt(x) + α, where α ∈ Rn is a constant vector independent of I.

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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) ≦ 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) ≦ c · opt(x) + α, where α ∈ Rn is a constant vector independent of I.

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Bi-Objective Online Search

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q q⋆ p⋆

for t = 1, . . . do Accept a request rt if pt ≥ p⋆or qt ≥ q⋆ end

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q q⋆ p⋆

for t = 1, . . . do Accept a request rt if pt ≥ p⋆or qt ≥ q⋆ end c = max P p , Q q⋆ , P p⋆ , Q q

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q q⋆ p⋆

for t = 1, . . . do Accept a request rt if pt ≥ p⋆and qt ≥ q⋆ end

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q q⋆ p⋆

for t = 1, . . . do Accept a request rt if pt ≥ p⋆and qt ≥ q⋆ end c = max P p , q⋆ q , p⋆ p , Q q

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q

z⋆ Q z⋆ q

q⋆ p⋆ pt · qt = z⋆

for t = 1, . . . do Accept a request rt if pt · qt ≥ z⋆ end

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Bi-Objective Online Search

◮ request rt =

• pt, qt

⊺ in each time period t = 1, . . . , T

◮ pt ∈ [p, P] where 0 < p ≤ P, and qt ∈ [q, Q] where 0 < q ≤ Q

Reservation Price Policy

pt qt p P q Q

z⋆ Q z⋆ q

q⋆ p⋆ pt · qt = z⋆

for t = 1, . . . do Accept a request rt if pt · qt ≥ z⋆ end c =

• PQ

pq

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Randomization

p1

t

p2

t

m120 m121 m122 m123 m124 m125 = M1 m220 m221 m222 m223 = M2 · · · p1

t · p2 t = m126

p1

t · p2 t = m227 bc bc bc bc bc bc bc

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Conclusion & Further Research

b b b b b b b

√Pp P p

b b b

• max
• profit

appreciation

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Conclusion & Further Research

b b b b b b b

√Pp P p

b b b

• max
• profit

appreciation

• ◮ application to multi-objective versions of classical online problems

◮ relations between single- and multi-objective online optimization ◮ alternative concepts

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The Multi-Objective k-Canadian Traveller Problem

s t 1

• 1