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

b b b Who gets your antique car? 2/12

b b b Who gets your antique car? � � profit max appreciation � 2/12

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. 3/12

b b b b b b b Online Optimization (contd.) P for t = 1 , . . . , T do Accept a price p t if � p t ≥ Pp end √ Pp � P � alg is p -competitive. p 4/12

Multi-Objective Optimization Consider a multi-objective optimization problem P for a given feasible set X ⊆ R n , and objective vector f : X �→ R k : max f ( x ) P s.t. x ∈ X 5/12

Multi-Objective Optimization Consider a multi-objective optimization problem P for a given feasible set X ⊆ R n , and objective vector f : X �→ R k : max f ( x ) P 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 ∈ R n , x � y : ⇔ x i ≤ y i , for i = 1 , . . . , n , and x � = y . 5/12

Multi-Objective Optimization Consider a multi-objective optimization problem P for a given feasible set X ⊆ R n , and objective vector f : X �→ R k : max f ( x ) P 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 ∈ R n , x � y : ⇔ x i ≤ y i , for i = 1 , . . . , n , and x � = y . ◮ If ˆ x is an efficient solution, f (ˆ x ) is called non-dominated point. 5/12

b b b Multi-Objective Optimization (contd.) � � profit max appreciation � 6/12

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 ◮ set of inputs I 7/12

Multi-Objective Online Problem Multi-objective (online) optimization problem P ◮ set of inputs I ◮ set of feasible outputs X ( I ) ∈ R n for I ∈ I 7/12

Multi-Objective Online Problem Multi-objective (online) optimization problem P ◮ set of inputs I ◮ set of feasible outputs X ( I ) ∈ R n for I ∈ I ◮ objective function f given as f : I × X �→ R n + 7/12

Multi-Objective Online Problem Multi-objective (online) optimization problem P ◮ set of inputs I ◮ set of feasible outputs X ( I ) ∈ R n for I ∈ I ◮ objective function f given as f : I × X �→ R n + ◮ algorithm alg ◮ feasible solution alg [ I ] ∈ X ( I ) ◮ associated objective alg ( I ) = f ( I , alg [ I ]) 7/12

Multi-Objective Online Problem Multi-objective (online) optimization problem P ◮ set of inputs I ◮ set of feasible outputs X ( I ) ∈ R n for I ∈ I ◮ objective function f given as f : I × X �→ R n + ◮ 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 ) 7/12

Multi-Objective Approximation Algorithms 8/12

Multi-Objective Approximation Algorithms ρ -approximation of a solution x f i ( x ′ ) ≤ ρ · f i ( x ) for i = 1 , . . . , n 8/12

Multi-Objective Approximation Algorithms ρ -approximation of a solution x f i ( x ′ ) ≤ ρ · f i ( 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 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 ) ≦ c · opt ( x ) + α , where α ∈ R n 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 ) ≦ c · opt ( x ) + α , where α ∈ R n 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 ∈ R n is a constant vector independent of I . 9/12

Bi-Objective Online Search 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy q t Q for t = 1 , . . . do Accept a request r t if p t ≥ p ⋆ or q t ≥ q ⋆ end q ⋆ q p t p p ⋆ P 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy q t Q for t = 1 , . . . do Accept a request r t if p t ≥ p ⋆ or q t ≥ q ⋆ end q ⋆ � P � p , Q q ⋆ , P p ⋆ , Q q c = max q p t p p ⋆ P 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy q t Q for t = 1 , . . . do Accept a request r t if p t ≥ p ⋆ and q t ≥ q ⋆ end q ⋆ q p t p p ⋆ P 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy q t Q for t = 1 , . . . do Accept a request r t if p t ≥ p ⋆ and q t ≥ q ⋆ end q ⋆ � P � p , q ⋆ q , p ⋆ p , Q q c = max q p t p p ⋆ P 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy p t · q t = z ⋆ q t Q for t = 1 , . . . do Accept a request r t if p t · q t ≥ z ⋆ q ⋆ end q p t p p ⋆ z ⋆ z ⋆ P Q q 10/12

Bi-Objective Online Search � ⊺ in each time period t = 1 , . . . , T ◮ request r t = � p t , q t ◮ p t ∈ [ p , P ] where 0 < p ≤ P , and q t ∈ [ q , Q ] where 0 < q ≤ Q Reservation Price Policy p t · q t = z ⋆ q t Q for t = 1 , . . . do Accept a request r t if p t · q t ≥ z ⋆ q ⋆ end � PQ q c = pq p t p p ⋆ z ⋆ z ⋆ P Q q 10/12

bc bc bc bc bc bc bc Randomization p 2 t m 2 2 3 = M 2 · · · p 1 t · p 2 t = m 1 2 6 p 1 t · p 2 t = m 2 2 7 m 2 2 2 m 2 2 1 m 2 2 0 m 1 2 0 m 1 2 1 m 1 2 2 m 1 2 5 = M 1 m 1 2 3 m 1 2 4 p 1 t 11/12

b b b b b b b b b b Conclusion & Further Research � � profit max appreciation � P √ Pp p 12/12

b b b b b b b b b b Conclusion & Further Research � � profit max appreciation � P √ Pp p ◮ application to multi-objective versions of classical online problems ◮ relations between single- and multi-objective online optimization ◮ alternative concepts 12/12

The Multi-Objective k -Canadian Traveller Problem � 1 � � 0 � 0 0 s t � 0 � � 0 � 1 0 13/12