Dynamic programming bi-criteria combinatorial optimization

Dynamic Programming Multi-Objective Combinatorial Optimization

“…I 3 Set of Pareto optimal points for a bi-criteria optimization of parenthesizations relative to Ψ (1) and Ψ (2) . Matrices dimensions are from the interval [1,10], n = 5.4 Set of Pareto optimal points for a bi-criteria optimization of parenthesizations relative to Ψ (1) and Ψ (3) . Matrices dimensions are from the interval [1,10], n = 6.4 Set of Pareto optimal points for a bi-criteria optimization of global sequence alignments relative to Ψ (1) and Ψ (2) for n = 50 and k = 4 .…”

“…. 12.4 Set of Pareto optimal points for a bi-criteria optimization of bitonic tours (n = 300, x max = 1000 and y max = 3000) relative to Ψ (1) and Ψ (2) 12.5 Set of Pareto optimal points for a bi-criteria optimization of bitonic tours (n = 300, x max = 1000 and y max = 3000) relative to Ψ (1) and Ψ (3) 13.1 Sequence of points p 1 , p 2 , p 3 , p…”

“…5.1 Average number of parenthesizations described by the circuit before optimization ∅, after optimization relative to Ψ (1) , relative to Ψ (2) , and relative to Ψ (3) with matrices dimensions from the intervals [1,10] or [1,100] among 100 trials .…”

“…2 Average number of parenthesizations described by the circuit after optimization relative to Ψ (1) , Ψ (2) , relative to Ψ (2) , Ψ (1) , relative to Ψ (1) , Ψ (3) , relative to Ψ (3) , Ψ (1) , relative to Ψ (2) , Ψ (3) , and relative to Ψ (3) , Ψ (2) with matrices dimensions from the interval [1,10] among 100 trials . 5.3 Average value of Ψ (1) after optimization relative to Ψ (1) and relative to Ψ (3) , Ψ (1) , and average value of Ψ (3) after optimization relative to Ψ (3) , relative to Ψ (1) , Ψ (3) , and relative to Ψ (2) , Ψ (3) with matrices dimensions from the interval [1,10] among 100 trials . .…”

“…Matrices dimensions are from the interval [1, k] 5.6 Number of cases with totally optimal parenthesizations relative to Ψ (1) and Ψ (2) among 100 trials. Matrices dimensions are from the interval 5.7 Number of cases with totally optimal parenthesizations relative to Ψ (1) and Ψ (3) among 100 trials. Matrices dimensions are from the interval…”