A Quadratic Edge-Finding Filtering Algorithm for Cumulative Resource - PowerPoint PPT Presentation

Introduction The Edge-finding Experimental results Conclusion and Perspectives A Quadratic Edge-Finding Filtering Algorithm for Cumulative Resource Constraints Roger Kameugne 1 , 2 Laure Pauline Fotso 2 Joseph Scott 3 Youcheu Ngo-Kateu 2 1 University of Maroua, Dept. of Mathematics, Maroua-Cameroon 2 University of Yaound´ e I, Dept. of Mathematics and Computer Sciences, Yaound´ e-Cameroon 3 Uppsala University, Computing Science Division, Uppsala Sweden Perugia-Italy CP 2011 Perugia-Italy R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Outline 1 Introduction 2 The Edge-finding Edge-finding rule Quadratic edge-finding algorithm 3 Experimental results 4 Conclusion and Perspectives R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Outline 1 Introduction 2 The Edge-finding Edge-finding rule Quadratic edge-finding algorithm 3 Experimental results 4 Conclusion and Perspectives R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Overview 1 Edge-finding is one of the most important and successful filtering algorithms used in constraint-based scheduling. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Overview 1 Edge-finding is one of the most important and successful filtering algorithms used in constraint-based scheduling. 2 It is well-understood for disjunctive scheduling problems. There exist efficient algorithms running in time O ( n log n ) where n is the number of tasks on the resource. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Overview 1 Edge-finding is one of the most important and successful filtering algorithms used in constraint-based scheduling. 2 It is well-understood for disjunctive scheduling problems. There exist efficient algorithms running in time O ( n log n ) where n is the number of tasks on the resource. 3 For cumulative scheduling, edge-finding is more challenging since task may require several capacity units. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Overview 1 Edge-finding is one of the most important and successful filtering algorithms used in constraint-based scheduling. 2 It is well-understood for disjunctive scheduling problems. There exist efficient algorithms running in time O ( n log n ) where n is the number of tasks on the resource. 3 For cumulative scheduling, edge-finding is more challenging since task may require several capacity units. 4 I will present a sound, quadratic edge-finding algorithm for cumulative resource constraints. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Overview 1 Edge-finding is one of the most important and successful filtering algorithms used in constraint-based scheduling. 2 It is well-understood for disjunctive scheduling problems. There exist efficient algorithms running in time O ( n log n ) where n is the number of tasks on the resource. 3 For cumulative scheduling, edge-finding is more challenging since task may require several capacity units. 4 I will present a sound, quadratic edge-finding algorithm for cumulative resource constraints. 5 Experimental results on benchmarks from the Project Scheduling Problem Library suggest that the new algorithm is faster in practice. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Related works 1 It is proved in [Mercier and Van Hentenryck, 2008], that the O ( kn 2 ) edge-finding algorithm of [Nuijten 1994] and it refine version running in O ( n 2 ) ([Baptiste, et al., 2001]) are incomplete: They do not perform all the edge-finding updates. Mercier and Van Hentenryck then decide to fixe the first one and propose a complete algorithm running in O ( kn 2 ). 2 There is an edge finding algorithm with time complexity O ( kn log n ) in [Vil´ ım, 2009]. 3 Recently, a sound and quadratic edge-finding algorithm based on minimum capacity profil was described in [Vil´ ım, 2011]. R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems 1 a set T of tasks sharing a resource of capacity C 2 Each task i ∈ T must be executed without interruption over p i units of time between an earliest start time r i and a latest end time d i 3 each task i ∈ T requires a constant amount of resource c i . time e i = c i p i C p i 0 5 10 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems 1 a set T of tasks sharing a resource of capacity C 2 Each task i ∈ T must be executed without interruption over p i units of time between an earliest start time r i and a latest end time d i 3 each task i ∈ T requires a constant amount of resource c i . r i =1 d i =9 e i = c i p i C p i 0 5 10 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems 1 a set T of tasks sharing a resource of capacity C 2 Each task i ∈ T must be executed without interruption over p i units of time between an earliest start time r i and a latest end time d i 3 each task i ∈ T requires a constant amount of resource c i . e i = c i p i c i C p i 0 5 10 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems A solution of a CuSP is a schedule that assigns a starting date s i to each task i such that: ∀ i ∈ T : r i ≤ s i ≤ s i + p i ≤ d i (1) � ∀ τ : c i ≤ C (2) i ∈ T , s i ≤ τ< s i + p i 1 (1) ensures that each task is assigned a feasible start and end time 2 (2) enforces the resource constraint The Cumulative Scheduling Problem is NP-Complete R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems Extension of release date, deadline and energy notations from tasks to sets of tasks: � r Ω = min j ∈ Ω r j , d Ω = max j ∈ Ω d j , e Ω = e j (3) j ∈ Ω r { D , A } =1 r { B , C } =4 D A B C 0 3 6 9 12 15 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Experimental results Conclusion and Perspectives Cumulative Scheduling Problems Extension of release date, deadline and energy notations from tasks to sets of tasks: � r Ω = min j ∈ Ω r j , d Ω = max j ∈ Ω d j , e Ω = e j (3) j ∈ Ω d { A , B , C } =8 d D =15 D A B C 0 3 6 9 12 15 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Edge-finding rule Experimental results Quadratic edge-finding algorithm Conclusion and Perspectives Outline 1 Introduction 2 The Edge-finding Edge-finding rule Quadratic edge-finding algorithm 3 Experimental results 4 Conclusion and Perspectives R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Edge-finding rule Experimental results Quadratic edge-finding algorithm Conclusion and Perspectives Edge-finding rule: detection Let us consider the set of tasks Ω = { A , B , C } . No matter how we arrange the tasks in Ω ∪ { D } , D cannot be completed before t = 8, but all tasks in Ω = { A , B , C } must end before t = 8; r Ω ∪{ D } =1 d Ω =8 d D =15 D A B C 0 5 10 15 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative

Introduction The Edge-finding Edge-finding rule Experimental results Quadratic edge-finding algorithm Conclusion and Perspectives Edge-finding rule: detection Let us consider the set of tasks Ω = { A , B , C } . No matter how we arrange the tasks in Ω ∪ { D } , D cannot be completed before t = 8, but all tasks in Ω = { A , B , C } must end before t = 8; therefore, D must end after the end of all tasks in Ω. r Ω ∪{ D } =1 d Ω =8 d D =15 D Ω 0 5 10 15 R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu Quadratic Edge-Finding Algorithm for Cumulative