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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

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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 Scott3 Youcheu Ngo-Kateu 2

1University of Maroua, Dept. of Mathematics, Maroua-Cameroon 2University of Yaound´

e I, Dept. of Mathematics and Computer Sciences, Yaound´ e-Cameroon

3Uppsala 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

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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

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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

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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

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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

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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

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SLIDE 7

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

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SLIDE 8

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

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

Related works

1 It is proved in [Mercier and Van Hentenryck, 2008], that the

O(kn2) edge-finding algorithm of [Nuijten 1994] and it refine version running in O(n2) ([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(kn2).

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

  • n 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

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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

pi units of time between an earliest start time ri and a latest end time di

3 each task i ∈ T requires a constant amount of resource ci.

5 10

time C pi

ei = cipi

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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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

pi units of time between an earliest start time ri and a latest end time di

3 each task i ∈ T requires a constant amount of resource ci.

5 10

C pi

ei = cipi

ri=1 di=9

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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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

pi units of time between an earliest start time ri and a latest end time di

3 each task i ∈ T requires a constant amount of resource ci.

5 10

C pi

ei = cipi

ci

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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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 si to each task i such that: ∀i ∈ T : ri ≤ si ≤ si + pi ≤ di (1) ∀τ :

  • i∈T, si≤τ<si+pi

ci ≤ C (2)

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

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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: rΩ = min

j∈Ω rj,

dΩ = max

j∈Ω dj,

eΩ =

  • j∈Ω

ej (3)

3 6 9 12 15

A B C D

r{B,C}=4 r{D,A}=1

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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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: rΩ = min

j∈Ω rj,

dΩ = max

j∈Ω dj,

eΩ =

  • j∈Ω

ej (3)

3 6 9 12 15

A B C D

d{A,B,C}=8 dD=15

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

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

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

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; 5 10 15

A B C D

rΩ∪{D}=1 dΩ=8 dD=15

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

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. 5 10 15

Ω

D

rΩ∪{D}=1 dΩ=8 dD=15

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

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.

eΩ∪{D} > C · (dΩ − rΩ∪{D}) = ⇒ Ω ⋖ D (4)

5 10 15

Ω

D

rΩ∪{D}=1 dΩ=8 dD=15

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Edge-finding rule: adjustment

1 If Ω ⋖ D then Θ ⋖ D for all Θ ⊆ Ω 2 The adjustment phase only considers sets of tasks Θ having

enough energy to affect the schedule of D For Θ = Ω = {A, B, C}

5 10 15

dΘ − rΘ C − cD (C − cD)(dΘ − rΘ) rest rΘ=1 dΘ=8

D

LBD ≥ rΘ +

  • eΘ−(C−cD)(dΘ−rΘ)

cD

  • = 3
  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Edge-finding rule: adjustment

1 If Ω ⋖ D then Θ ⋖ D for all Θ ⊆ Ω 2 The adjustment phase only considers sets of tasks Θ having

enough energy to affect the schedule of D For Θ = Ω = {A, B, C} For Θ = {B, C}

5 10 15

dΘ − rΘ C − cD rest rΘ=4 dΘ=8

D

LBD ≥ rΘ + ⌈ 1

cD rest⌉ = 6

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Edge-finding rule

LBi = max     ri, max

Ω⊆T i / ∈Ω α(Ω,i)

max

Θ⊆Ω rest(Θ,ci)>0

rΘ + rest (Θ, ci) ci

    (5) with α (Ω, i) def =

  • C
  • dΩ − rΩ∪{i}
  • < eΩ∪{i}
  • ∨ (ri + pi ≥ dΩ)

(6) rest(Θ, ci) = eΘ − (C − ci) (dΘ − rΘ) if Θ = ∅

  • therwise

(7)

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Edge-finding rule: dominance properties

1 It is demonstrated in [Mercier and Van Hentenryck, 2008]

that an edge-finder that only considers sets Ω ⊆ T and Θ ⊆ Ω which are also task intervals can be complete.

2 The task interval ΩL,U is the set of tasks

ΩL,U = {j ∈ T | rL ≤ rj ∧ dj ≤ dU} (8)

3 If the edge-finding condition detects Ω ⋖ i then there exists a

task L ∈ T for which rL ≤ ri and ΩL,U ⋖ i with dΩ = dU.

4 If the edge-finding condition detects Ω ⋖ i then dΩ < di.

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

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

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Slack

1 If C

  • dΩ − rΩ∪{i}
  • < eΩ∪{i} then there exists a task L ∈ T for

which rL ≤ ri and ei > C (dU − rL) − eΩL,U with dΩ = dU.

2 The slack of the task set Ω, denoted SLΩ, is the free energy

  • f the interval defined by this set of tasks.

SLΩ = C(dΩ − rΩ) − eΩ

5 5

A B C

rA,B,C=1 dA,B,C=8 SL{A,B,C} = 5

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Minimum slack

1 Let U and i be two tasks. There is a task τ(U, i) which

defines the lower bound of the task intervals of minimum slack among ΩL,U with rL ≤ ri i.e., rτ(U,i) ≤ ri and ∀L ∈ T, rL ≤ ri : SLΩτ(U,i),U ≤ SLΩL,U (9)

2 For tasks L, U ∈ T and task i /

∈ ΩL,U ΩL,U ⋖ i ⇐ ⇒ ei > SLΩτ(U,i),U ∧ dU < di (10)

3 If Θl,u is the interval responsible for the maximum adjustment

  • f ri with rl ≤ ri, then:

rτ(u,i) +

  • rest
  • Θτ(u,i),u, ci
  • ci
  • ≥ rl +

rest (Θl,u, ci) ci

  • (11)
  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Density

1 When Ω ⋖ i is detected, a set of tasks Θ ⊆ Ω is used to

update ri if rest(Θ, ci) > 0.

2 eΘ − (C − ci)(dΘ − rΘ) > 0 ⇐

⇒

eΘ dΘ−rΘ > C − ci

3 The density of the task set Θ, denoted DensΘ, is given by

DensΘ =

eΘ dΘ−rΘ .

5

DensΘ rΘ=1 dΘ=8 DensΘ ≈ 2.28

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Maximum Density

1 Let u and i be two tasks. There exists a task ρ(u, i) which

defines the lower bound of the task intervals of maximum density among Θl,u with ri < rl i.e., ri < rρ(u,i) and ∀l ∈ T, ri < rl : DensΘρ(u,i),u ≥ DensΘl,u (12)

2 If Θl,u ⊆ Ω is the interval responsible for the maximum ri, and

ri < rl, then:        rρ(u,i) +

  • rest(Θρ(u,i),u,ci)

ci

  • ≥ rl +
  • rest(Θl,u,ci)

ci

  • if ri < rl ≤ rρ(u,i)

rρ(u,i) +

  • rest(Θρ(u,i),u,ci)

ci

  • > ri

if ri < rρ(u,i) < rl (13)

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

The quadratic Edge-finder algorithm

for U ∈ T by increasing dU do for i ∈ T by decreasing ri do if di ≤ dU then if DensΩi,U > DensΩρ,U then ρ ← i; else Dupdi ← potentiel update value using Ωρ,U; for i ∈ T by increasing ri do if SLΩi,U < SLΩτ,U then τ ← i; if di > dU then SLupdi ← potentiel update value using Ωτ,U; if Ωτ,U ⋖ i then LBi ← max(ri, Dupdi, SLupdi);

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives Edge-finding rule Quadratic edge-finding algorithm

Algorithm properties

1 The algorithm presented here is quadratic and correctly

detects all Ω ⋖ i.

2 This algorithm is sound and reaches the same fixpoint as

previous egde-finding algorithm.

3 In most cases, the algorithm performs the maximum

adjustment at the first run, but its worst case complexity for the maximum adjustment is O(n3).

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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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

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

Experimental Results

Implemented in C++ using the Gecode 3.4.2

Existing cumulative propagator which contains a sequence of three filters: Θ-tree edge-finding, overload checking and time tabling. Modified version that substituted the new quadratic edge-finding filter for the Θ-tree filter .

Benchmarks from the PSPLib single-mode sets: 480 instances each of 30, 60 and 90 tasks. With the time limits of 300 sec, 1034 instances was solved by both filters 8 instances were solved only by the quadratic propagator in the time available.

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

Runtime Comparison

50 100 150 200 250 300 50 100 150 200 250 300 Θ-tree runtimes (sec) Quadratic runtimes (sec) j30 j60 j90

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

Propagations Comparison

10 100 1000 10000 100000 1e+06 1e+07 10 100 1000 10000 100000 1e+06 1e+07 1e+08 Θ-tree EF Quadratic EF j30 j60 j90

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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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

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

Conclusion and Perspectives

Conclusion:

Using a combination of minimum slack and maximum density, cumulative edge-finding can be performed in O(n2) time. Reaches same fixpoint as a complete edge-finder, but may take additional iterations to do so. While not strictly dominating O(kn log n) Θ-tree edge-finding, experimental results suggest that the quadratic edge-finding is faster in practice.

Perspectives:

Extended the empirical evaluation of the algorithm Extended the following approach to the extended edge-finding (in preparation) Analysed the possibily to use the Θ-tree to reduce the complexity of this algorithm

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

I thank

1 The AFPC and the ACP for their grant support to the first

author;

2 Christian Schulte, for providing a patch for the Gecode

cumulative propagator;

3 The Swedish Research Council, for their grant support to the

third author;

4 all of you for your attention today.

Questions?

  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

References

  • P. Baptiste, C. Le Pape, and W.P.M. Nuijten

Constraint-based scheduling: applying constraint programming to scheduling problems Kluwer, 2001

  • L. Mercier and P. Van Hentenryck

Edge finding for cumulative scheduling INFORMS Journal on Computing, 20:143–153, 2008

  • P. Vil´

ım Edge finding filtering algorithm for discrete cumulative resources in O(kn log n) In I.P. Gent, ed., CP2009 (LCNS 5732) , pp. 802–816, 2009. Springer-Verlag

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Introduction The Edge-finding Experimental results Conclusion and Perspectives

References

  • P. Vil´

ım Timetable Edge Finding Filtering Algorithm for Discrete Cumulative Resources. In Proceedings of CPAIOR’2011. pp.230 245.

  • W. Nuijten

Time and resource constrained scheduling: a constraint satisfaction approach. PhD thesis, Eindhoven University of

  • Technology. (1994).
  • R. Kameugne, L. P. Fotso, J. Scott, Y. Ngo-Kateu

Quadratic Edge-Finding Algorithm for Cumulative