A Quadratic Edge-Finding Algorithm
for Cumulative Resource Constraints
R. Kameugne1,2
L.P. Fotso2
Y. Ngo-Kateu2
1
2
J. Scott3
Department of Mathematics
University of Maroua
Departments of Mathematics and Computer Science
University of Yaoundé
3
ASTRA Group, Computing Science Division
University of Uppsala
SweConsNet, 2011
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Overview
Edge-finding is a filtering algorithm used in constraint-based
scheduling applications
For disjunctive scheduling, there are edge-finding algorithms
with complexity of (𝑛 log 𝑛).
For cumulative edge-finding, the best algorithm to this point was
(𝑘𝑛 log 𝑛) [Vilím, 2009].
Today I will present an algorithm for cumulative edge-finding in
𝑛2 time.
• Complexity doesn’t strictly dominate previous algorithms, but
offers a practical improvement.
2. SweConsNet | Quadratic Edge-Finding
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Outline
1. Introduction: Cumulative Scheduling
2. Edge-Finding
3. A Quadratic Algorithm
Minimum Slack and Maximum Density
Filtering Algorithm
4. Experimental Results
3. SweConsNet | Quadratic Edge-Finding
Results
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
The cumulative resource scheduling problem
Set of tasks 𝑇 , sharing a finite resource of constant capacity .
Inelastic: task has fixed resource requirement 𝑐 and duration 𝑝
• Energy of the task is 𝑒 = 𝑐 ⋅ 𝑝
Non-preemptive: task may not be interrupted once started.
A task is bounded by earliest and latest start times, est and lst .
• With fixed 𝑝, can reason on last completion time lct instead.
est 𝑖 = 0
lst 𝑖 = 7
𝑐𝑖
4. SweConsNet | Quadratic Edge-Finding
𝑒𝑖
𝑝𝑖
.
0
lct 𝑖 = 10
5
10
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
The cumulative resource scheduling problem
Raise several of these concepts to apply to a set of tasks Ω:
est Ω = min est 𝑖
lct Ω = max lct 𝑖
𝑖∈Ω
𝑖∈Ω
eΩ =
e𝑖
𝑖∈Ω
est 𝐴,𝐷 = 0
lct 𝐴,𝐵,𝐶 = 5
𝐶
lct 𝐷 = 10
𝐷
𝐴
𝐵
.
0
est 𝐵,𝐶 = 2
example due to [Vilím, 2009]
5. SweConsNet | Quadratic Edge-Finding
5
10
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
The cumulative constraint
Need to find a start time for each task, so that at any given time
the tasks that are executing do not require more of the
resource than is available.
The cumulative constraint enforces the rule:
∀𝑡
𝑐𝑖 ≤
𝑖∈𝑇
𝑠𝑖 ≤𝑡≤𝑠𝑖 +𝑝𝑖
Will only enforce bounds consistency
• Try to tighten earliest and latest start times
• Not concerned with infeasible start times inside the domain
cumulative is NP-complete
• Polynomial time filtering algorithms for cumulative, such as
edge-finding, apply relaxations of the constraint
6. SweConsNet | Quadratic Edge-Finding
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Edge-finding detects required ordering
Consider the set of tasks Ω = {𝐴, 𝐵, 𝐶}.
Is there a feasible schedule in which Ω and 𝐷 both end by lct Ω ?
• If not, 𝐷 must end after the end of every task in Ω.
eΩ∪{𝐷} >
⋅ (lct Ω − est Ω∪{𝐷} ) ⟹ Ω ⋖ 𝐷
est ∗𝐷 = 3
est Ω = 0
𝐶
lct Ω = 5
lct 𝐷 = 10
𝐷
𝐴
.
0
8. SweConsNet | Quadratic Edge-Finding
𝐵
5
10
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
How much can the bound be updated?
Consider eΩ in two parts:
• energy which may be scheduled without affecting 𝐷,
• and the rest , which must affect 𝐷.
est Ω = 0
rest(Ω, 𝑐𝐷 )
.
𝐷
𝐷
est ∗𝐷 = 3
0
rest(Ω, 𝑐𝑖 ) =
10
5
eΩ −( − 𝑐𝑖 )(lct Ω − est Ω )
0
est ∗𝑖 ≥ est Ω +
9. SweConsNet | Quadratic Edge-Finding
lct 𝐷 = 10
lct Ω = 5
rest(Ω, 𝑐𝑖 )
𝑐𝑖
if Ω ≠ ∅
otherwise
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Must consider subsets of Ω
{𝐵, 𝐶} not energetic enough to require {𝐵, 𝐶} ⋖ 𝐷.
{𝐴, 𝐵, 𝐶} ⋖ 𝐷, and lct 𝐴,𝐵,𝐶 ≥ lct 𝐵,𝐶 ,
• therefore 𝐷 must end after lct 𝐵,𝐶 .
est 𝐷 = 0 est 𝐵,𝐶 = 2
lct 𝐵,𝐶 = 5
lct 𝐷 = 10
𝐷
.
0
est ∗𝐷 = 4
5
Must find Θ ⊆ Ω that yields the strongest update.
10. SweConsNet | Quadratic Edge-Finding
10
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
The Edge-Finding Rule
⎛
⎜
rest Θ, 𝑐𝑖
est ∗𝑖 = max ⎜est 𝑖 , max max est Θ +
Θ⊆Ω
Ω⊆𝑇
⎜
𝑐𝑖
𝑖∉Ω rest Θ,𝑐𝑖 >0
⎜
𝛼(Ω,𝑖)
⎝
⎞
⎟
⎟
⎟
⎟
⎠
with
𝑑𝑒𝑓
𝛼 (Ω, 𝑖) =
lct Ω − est Ω∪{𝑖} < 𝑒Ω∪{𝑖} ∨ est 𝑖 +𝑝𝑖 ≥ lct Ω
rest Θ, 𝑐𝑖 =
𝑒Θ −
0
− 𝑐𝑖
lct Θ − est Θ
if Θ ≠ ∅
otherwise
(The rule for adjusting lct is symmetric.)
11. SweConsNet | Quadratic Edge-Finding
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Task Intervals
An edge-finder only needs to consider task intervals:
all the tasks that must be scheduled within a fixed window.
est 𝐴,𝐵,𝐶 = 0
est 𝐴,𝐵 = 0
lct 𝐴,𝐵,𝐶 = 5
lct 𝐴,𝐵 = 5
𝐶
𝐴
𝐴
𝐵
.
0
.
5
0
Defined by upper and lower bound tasks:
Ω𝑈
𝐿 = 𝑖 ∈ 𝑇 ∣ est 𝑖 ≥ est 𝐿 ∧ lct 𝑖 ≤ lct 𝑈
To update est 𝑖 , only need to consider:
• Ω𝑈
𝐿 such that lct 𝑈 < lct 𝑖
• Θ𝑢𝑙 such that est 𝐿 ≤ est 𝑙 ≤ lct 𝑢 ≤ lct 𝑈
12. SweConsNet | Quadratic Edge-Finding
𝐵
5
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Complexity
Naive: compare all tasks with all pairs of intervals.
•
𝑛5
Precompute potential updates outside
the precedence detection loop.
𝑛3
• Can reduce to
𝑘𝑛2 , where 𝑘 is the number of distinct
resource requirements [Mercier and Van Hentenryck, 2008].
•
Θ-tree: use a binary tree to compute
one bound of each interval [Vilím, 2009].
•
(𝑘𝑛 log 𝑛)
Quadratic
• Previous
incomplete.
13. SweConsNet | Quadratic Edge-Finding
𝑛2 algorithm [Baptiste, et al., 2001] was
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Definition: Slack
Slack is a measure of the capacity that is not required by the
tasks of an interval Ω:
𝑆𝐿(Ω) = (lct Ω − est Ω ) − 𝑒Ω
est 𝐴,𝐵,𝐶 = 0
lct 𝐴,𝐵,𝐶 = 5
𝐶
𝑆𝐿
𝑆𝐿({𝐴, 𝐵, 𝐶}) = 5
𝐴
.
0
15. SweConsNet | Quadratic Edge-Finding
𝐵
5
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Minimum Slack
For an upper bound 𝑈 and a task 𝑖, there is a lower bound
𝜏(𝑈, 𝑖) with est 𝜏(𝑈,𝑖) ≤ est 𝑖 which defines the interval of
minimum slack:
𝑈
∀𝐿 ∈ 𝑇, est 𝐿 ≤ est 𝑖 ∶ 𝑆𝐿 Ω𝑈
𝜏(𝑈,𝑖) ≤ 𝑆𝐿 Ω𝐿
For an Ω ⊂ 𝑇 and task 𝑖 ∉ Ω
Ω ⋖ 𝑖 ⟺ e𝑖 > 𝑆𝐿 Ω𝑈
𝜏(𝑈,𝑖)
If Θ𝑢𝑙 ⊆ Ω is the interval responsible for the maximum est ∗𝑖 , and
est 𝑙 ≤ est 𝑖 , then:
est ∗𝑖
16. SweConsNet | Quadratic Edge-Finding
⎡ rest Θ𝑢𝜏(𝑢,𝑖) , 𝑐𝑖
= est 𝜏(𝑢,𝑖) + ⎢
𝑐𝑖
⎢
⎢
⎤
⎥
⎥
⎥
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Definition: Density
For a task interval Θ, the density of Θ is given by:
eΘ
𝐷𝑒𝑛𝑠(Θ) =
lct Θ − est Θ
est Θ = 0
lct Θ = 5
est Θ = 0
lct Θ = 5
𝐶
𝐴
.
0
17. SweConsNet | Quadratic Edge-Finding
𝐷𝑒𝑛𝑠(Θ)
𝐵
.
5
0
5
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Maximum Density
For an upper bound 𝑢 and a task 𝑖, there is a lower bound 𝜌(𝑢, 𝑖)
with est 𝑖 < est 𝜌(𝑢,𝑖) which defines the interval of maximum
density:
∀𝑙 ∈ 𝑇, est 𝑖 < est 𝑙 ∶ 𝐷𝑒𝑛𝑠 Θ𝑢𝜌(𝑢,𝑖) ≥ 𝐷𝑒𝑛𝑠 Θ𝑢𝑙
If Θ𝑢𝑙 ⊆ Ω is the interval responsible for the maximum est ∗𝑖 , and
est 𝑖 < est 𝑙 , then:
est ∗𝑖
18. SweConsNet | Quadratic Edge-Finding
⎡ rest Θ𝑢𝜌(𝑢,𝑖) , 𝑐𝑖
= est 𝜌(𝑢,𝑖) + ⎢
𝑐𝑖
⎢
⎢
⎤
⎥
⎥
⎥
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
for 𝑈 ∈ 𝑇 by increasing lct 𝑈 do
𝐸𝑛𝑒𝑟𝑔𝑦 ← 0, 𝑚𝑎𝑥𝐸𝑛𝑒𝑟𝑔𝑦 ← 0, est 𝜌 ← −∞, 𝑚𝑖𝑛𝑆𝐿 ← −∞
for 𝑖 ∈ 𝑇 by decreasing est 𝑖 do
if lct 𝑖 ≤ lct 𝑈 then
𝐸𝑛𝑒𝑟𝑔𝑦 ← 𝐸𝑛𝑒𝑟𝑔𝑦 + e𝑖
if lct𝐸𝑛𝑒𝑟𝑔𝑦
> lct𝑚𝑎𝑥𝐸
then
− est
− est
𝑢
𝑖
𝑈
𝜌
𝑚𝑎𝑥𝐸 ← 𝐸𝑛𝑒𝑟𝑔𝑦, est 𝜌 ← est 𝑖
else
𝐷𝑢𝑝𝑑𝑖 ← max 𝐷𝑢𝑝𝑑𝑖 , est 𝜌 +
𝑚𝑎𝑥𝐸−( −𝑐𝑖 )(lct 𝑈 − est 𝜌 )
𝑐𝑖
𝐸𝑖 ← 𝐸𝑛𝑒𝑟𝑔𝑦
for 𝑖 ∈ 𝑇 by increasing est 𝑖 do
if (lct 𝑈 − est 𝑖 ) − 𝐸𝑖 < 𝑚𝑖𝑛𝑆𝐿 then
est 𝜏 ← est 𝑖 , 𝑚𝑖𝑛𝑆𝐿 ← (lct 𝑈 − est 𝜏 ) − 𝐸𝑖
if lct 𝑖 > lct 𝑈 then
𝑟𝑒𝑠𝑡 ← 𝑐𝑖 (lct 𝑈 − est 𝜏 ) − 𝑚𝑖𝑛𝑆𝐿
if est 𝜏 ≤ lct 𝑈 ∧𝑟𝑒𝑠𝑡 > 0 then
𝑆𝐿𝑢𝑝𝑑𝑖 ← max 𝑆𝐿𝑢𝑝𝑑𝑖 , est 𝜏 +
if 𝑚𝑖𝑛𝑆𝐿 − e𝑖 < 0 then
𝐿𝐵 ← max(est 𝑖 , 𝐷𝑢𝑝𝑑𝑖 , 𝑆𝐿𝑢𝑝𝑑𝑖 )
20. SweConsNet | Quadratic Edge-Finding𝑖
𝑟𝑒𝑠𝑡
𝑐𝑖
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Completeness
The quadratic algorithm in [Baptiste, et al., 2001] is weaker than
edge-finding (misses some bound updates).
The algorithm presented here correctly detects all Ω ⋖ 𝑖.
Will not always make the strongest update on the first iteration,
but will make some update.
• Does make the strongest update if:
▶ est ≤ est , or
Θ
𝑖
▶ est < est ≤ est .
𝑖
Θ
𝜌
• Otherwise, requires 𝑛 − 1 iterations in the worst case.
“Lazy” evaluation, which makes sense for filtering algorithms.
21. SweConsNet | Quadratic Edge-Finding
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Experimental Results
Implemented in Gecode (3.4.2)
• Compared with existing implementation of the cumulative
Θ-tree filter
• Propagators executed each edge-finding filter in sequence with
time-tabling and overload checking.
Benchmarks from the PSPLib single-mode sets: 480 instances
each of 30, 60 and 90 tasks.
On average, less than 1% difference in number of propagations
required to find solution.
• For J60 and J90, the quadratic propagator ranged from <2%
fewer propagations to <5% more.
• For J30, one instance of 30% fewer, another of 15% more.
23. SweConsNet | Quadratic Edge-Finding
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Runtime Comparison
PSPLib Single-Mode RCSP benchmarks J30, J60, and J90
Quadratic runtimes (sec)
250
j30
j60
j90
200
150
100
50
0
0
50
100
150
Θ-tree runtimes (sec)
24. SweConsNet | Quadratic Edge-Finding
200
250
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Summary
Using a combination of minimum slack and maximum density,
cumulative edge-finding can be performed in
𝑛2 time.
Reaches same fixpoint as a complete edge-finder,
but may take additional iterations to find it.
While not strictly dominating (𝑘𝑛 log 𝑛) Θ-tree edge-finding,
experimental results suggest quadratic edge-finding is faster in
practice.
Further work:
•
𝑛2 extended edge-finding (in preparation)
• Is it possible to use a Θ-tree to reduce the complexity further?
25. SweConsNet | Quadratic Edge-Finding
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Thanks to. . .
The Swedish Research Council, as the fourth author is
supported by grant 2009-4384;
Christian Schulte, for providing a patch for the Gecode
cumulative propagator;
and last but not least: to all of you, for your attention today.
Questions?
26. SweConsNet | Quadratic Edge-Finding
Conclusion
Introduction
Edge-Finding
Quadratic EF
. . . . . .
Results
Conclusion
Further Reading
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. Vilím
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
27. SweConsNet | Quadratic Edge-Finding

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