Timetable Combinatorial Optimization David Adjiashvili , Sandro - PowerPoint PPT Presentation

Timetable Combinatorial Optimization David Adjiashvili , Sandro Bosio, Robert Weismantel, Rico Zenklusen IFOR, ETH Z¨ urich, Johns Hopkins University January 7, 2013

Motivation

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems.

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems. Complex Constraints Deal with complex constraints in dynamic setups.

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems. Complex Constraints Deal with complex constraints in dynamic setups. Existing theories

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems. Complex Constraints Deal with complex constraints in dynamic setups. Existing theories ◮ Scheduling

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems. Complex Constraints Deal with complex constraints in dynamic setups. Existing theories ◮ Scheduling ◮ Flow over time

Motivation Temporal Extensions Meaningful way to incorporate the dimension of time in Combinatorial Optimization problems. Complex Constraints Deal with complex constraints in dynamic setups. Existing theories ◮ Scheduling ◮ Flow over time ◮ · · ·

Timetable Combinatorial Optimization

Timetable Combinatorial Optimization Combinatorial Optimization (CO)

Timetable Combinatorial Optimization Combinatorial Optimization (CO) Input: ◮ Set system F = ( A, X ) (with X ⊂ 2 A ), ◮ Weights w a ∈ Z + for every a ∈ A .

Timetable Combinatorial Optimization Combinatorial Optimization (CO) Input: ◮ Set system F = ( A, X ) (with X ⊂ 2 A ), ◮ Weights w a ∈ Z + for every a ∈ A . Problem: Find S ∗ ∈ X maximizing w ( S ) = P a ∈ S w a .

Timetable Combinatorial Optimization Combinatorial Optimization (CO) Input: ◮ Set system F = ( A, X ) (with X ⊂ 2 A ), ◮ Weights w a ∈ Z + for every a ∈ A . Problem: Find S ∗ ∈ X maximizing w ( S ) = P a ∈ S w a . ⇓ Timetable Combinatorial Optimization (TCO)

Timetable Combinatorial Optimization Combinatorial Optimization (CO) Input: ◮ Set system F = ( A, X ) (with X ⊂ 2 A ), ◮ Weights w a ∈ Z + for every a ∈ A . Problem: Find S ∗ ∈ X maximizing w ( S ) = P a ∈ S w a . ⇓ Timetable Combinatorial Optimization (TCO) Input: ◮ A CO instance F = ( A, X ) , w . ◮ Durations τ a ∈ Z + for every a ∈ A . ◮ Total duration T ∈ Z + .

Timetable Combinatorial Optimization Combinatorial Optimization (CO) Input: ◮ Set system F = ( A, X ) (with X ⊂ 2 A ), ◮ Weights w a ∈ Z + for every a ∈ A . Problem: Find S ∗ ∈ X maximizing w ( S ) = P a ∈ S w a . ⇓ Timetable Combinatorial Optimization (TCO) Input: ◮ A CO instance F = ( A, X ) , w . ◮ Durations τ a ∈ Z + for every a ∈ A . ◮ Total duration T ∈ Z + . Problem: Find S 1 , · · · , S T ∈ X satisfying the duration property and maximizing w ( S 1 ) + · · · + w ( S T ) .

Timetable Combinatorial Optimization Timetable Combinatorial Optimization (TCO) Input: ◮ A CO instance F = ( A, X ) , w . ◮ Durations τ a ∈ Z + for every a ∈ A . ◮ Total duration T ∈ Z + . Problem: Find S 1 , · · · , S T ∈ X satisfying the duration property and maximizing w ( S 1 ) + · · · + w ( S T ) .

Timetable Combinatorial Optimization Timetable Combinatorial Optimization (TCO) Input: ◮ A CO instance F = ( A, X ) , w . ◮ Durations τ a ∈ Z + for every a ∈ A . ◮ Total duration T ∈ Z + . Problem: Find S 1 , · · · , S T ∈ X satisfying the duration property and maximizing w ( S 1 ) + · · · + w ( S T ) . duration property w.r.t. a ∈ A I a = { i ∈ [ T ] : a ∈ S i } is a disjoint union of intervals of length τ a .

Timetable Combinatorial Optimization Timetable Combinatorial Optimization (TCO) Input: ◮ A CO instance F = ( A, X ) , w . ◮ Durations τ a ∈ Z + for every a ∈ A . ◮ Total duration T ∈ Z + . Problem: Find S 1 , · · · , S T ∈ X satisfying the duration property and maximizing w ( S 1 ) + · · · + w ( S T ) . duration property w.r.t. a ∈ A I a = { i ∈ [ T ] : a ∈ S i } is a disjoint union of intervals of length τ a . duration property Duration property w.r.t a is satisfied for all a ∈ A .

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=8

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=13

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=17

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=13

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=17

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=21

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=21

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=22

TCO - Examples Example: Matchings ◮ A is the edge set of a graph G = ( V, A ) ◮ X is the set of all matchings of G Example: Timetable Matchings τ e w e τ e S 1 S 2 S 3 S 4 S 5 2 5 2 . 5 2 4 3 9 2 3 3 3 3 . 5 3 . 5 4 . 5 1 1 1 1 2 1 2 5 4 4 w ( S )=23

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A 8 9 6 10

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A

TCO - Examples Example: Integer Knapsack ◮ Let X = { S ⊆ A : | S | � 1 } , the rank-1 uniform matroid ◮ a ∈ X t if position t is occupied by an object of type a Knapsack ( T = 9) p 1 = 2 p 2 = 3 p 3 = 6 Groundset A

TCO - Examples Example: Orthogonal Retangle Packing ◮ X = { collections of intervals that do not overlap } .

TCO - Examples Example: Orthogonal Retangle Packing ◮ X = { collections of intervals that do not overlap } . w 1 = 3 w 2 = 5 w 3 = 2

TCO - Examples Example: Orthogonal Retangle Packing ◮ X = { collections of intervals that do not overlap } . w 1 = 3 w 2 = 5 w 3 = 2