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Course Scheduling in an Adjustable Constraint Propagation Schema Course Scheduling Nikolaos Pothitos, Panagiotis in an Adjustable Constraint Stamatopoulos, Kyriakos Zervoudakis Propagation Schema 1. Introduction 2. Related Work

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SLIDE 1

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling in an Adjustable Constraint Propagation Schema

Nikolaos Pothitos Panagiotis Stamatopoulos Kyriakos Zervoudakis

Department of Informatics and Telecommunications University of Athens

1

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Introduction

  • Course timetabling: an intensive problem.
  • Occurs in schools and academia.
  • Often still solved “by hand.”
  • Until utilizing Artificial Intelligence tools.
  • We design Constraint Programming methodologies.

2

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Related Work: Other Timetabling Solutions

  • Artificial Intelligence Paradigms

◮ Tabu Search ◮ Genetic Algorithms ◮ Local Search

  • Other Methodologies

◮ Network Flow ◮ Clustering to small sub-problems 3

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

We Chose Constraint Programming

  • Constraint Programming incorporates Artificial

Intelligence.

  • Also used in other fields.
  • Separates the problem declaration phase. . .
  • . . . from the solution search algorithm.
  • A flexible intelligent programming paradigm.

4

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Introduction to Constraint Satisfaction Problems

  • Constraint Programming solves Constraint Satisfaction

Problems (CSPs).

  • A CSP is described by:
  • 1. the variables of the problem,

◮ e.g. X, Y ,

  • 2. the domains of the variables,

◮ e.g. DX = {0, 1}, DY = {1, 3, 4},

  • 3. the constraints between the variables,

◮ e.g. X = Y . 5

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Solving a Constraint Satisfaction Problem

  • Assignment: makes a domain singleton.
  • Complete assignment: involves all variables.
  • Valid assignment: satisfies the constraints.
  • Solution: a valid complete assignment.

6

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Solving Procedure – Search Method

  • A posteriori constraint enforcement

◮ Checks constraints after the assignments.

  • Constraint Propagation

◮ Done a priori. ◮ Prunes no-good values from domains. 7

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Another CSP: N Queens

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8

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

9

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆

10

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SLIDE 11

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆

11

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆

12

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆

13

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆

14

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆

15

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆

16

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆

17

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆

18

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆

19

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆

20

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆

21

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SLIDE 22

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆

22

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆

23

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SLIDE 24

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆

24

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SLIDE 25

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆

25

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SLIDE 26

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆

26

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SLIDE 27

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆

27

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SLIDE 28

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆

28

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SLIDE 29

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆

29

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SLIDE 30

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆

30

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SLIDE 31

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆

31

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SLIDE 32

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆

32

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SLIDE 33

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆

33

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SLIDE 34

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆ ⋆

34

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SLIDE 35

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

35

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SLIDE 36

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

36

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

37

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SLIDE 38

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation for N Queens

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

38

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SLIDE 39

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation is Preventive

  • The search tree becomes limited.
  • But pruning costs.
  • Full constraint propagation is “expensive.”
  • A trade-off.

39

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Limiting Constraint Propagation

  • Propagate only important assignments

◮ E.g. big domain proportion removals

  • Limit the variables number affected

◮ Reduce the “domino” effect 40

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Our Contributions: Contents

  • 1. Course scheduling CSP formulation
  • 2. New semi-random heuristics
  • 3. New looser constraint propagation levels

41

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: Variables

  • Each lecture consists a variable

◮ xi: time-slot for lecture

  • Each lecture has its classroom

◮ cli: space-slot for lecture 42

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SLIDE 43

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: The “Resource” Constraint

  • Teachers and rooms are resources
  • Each teacher provides time-slots
  • At most one lecture/slot

43

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: A Teacher’s Constraints

x2

3

2 1

t1 x3 x1

4

Xt = {x1, x2, x3} , AllDifferent(Xt)

44

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SLIDE 45

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: Classrooms as Resources

  • Multiple rooms available for lectures
  • Two-dimensional time/room-slots

45

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: Constraint for Classrooms

r1 r2 r3

4

3

2 1

R (x3, cl3) (x4, cl4) (x5, cl5) (x2, cl2) (x1, cl1)

XR = {(x1, cl1), (x2, cl2), . . . , (x5, cl5)} , AllDifferent(XR)

46

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Course Scheduling: Constraints and Optimization

  • International Timetabling Competition 2007/08

specifications

  • More constraints and optimization criteria
  • Unification for various educational institutes

47

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Searching for a Solution

  • This phase is independent
  • Several methods can be employed

◮ Depth First Search (DFS) ◮ Depth-bounded Backtrack Search (DBS) ◮ Iterative Broadening ◮ . . . 48

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Stochastic Search Methods

  • Guided by random heuristics
  • A new hybrid semi-random heuristic
  • Gradually makes normal heuristics random
  • rand: the randomness degree level
  • More randomness while rand → 0+

49

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Normal Heuristics

  • Normal heuristics evaluate assignments.
  • Map values vi to hi.
  • Choose vi with maximum hi.

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Random Heuristics

  • Choose vi completely at random.
  • Corresponding probability Pi is same.
  • hi not taken into account.

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Normal + Random Heuristics = ?

  • Is there a compromise?

Pi = hrand

i

  • i hrand

i

  • hi is taken into account. . .
  • . . . especially as rand grows.

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Constraint Propagation: Less is More

  • We enforce limited constraint propagation
  • For variables with small domains
  • k: the domain size threshold
  • Domains with greater size excluded

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

k-Constraint-Propagation: Theoretical Motivation

  • d: The maximum domain size
  • Consistency between two variables: O(d2)
  • In our schema costs O(k · d)
  • k is less than d
  • Less consistency =

⇒ more flexibility

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SLIDE 55

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

k-Constraint-Propagation Speed

10 20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 14 100 200 300 400 Time (secs) k Dataset Time (secs)

  • We solve fourteen datasets
  • Real world course scheduling problems
  • k marginal values: worse results

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SLIDE 56

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

k-Constraint-Propagation Objective

10 20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 14 2000 4000 6000 Solution Cost k Dataset Solution Cost

  • Best k value: around 25
  • Slight solution quality improvement
  • But took less time

56

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SLIDE 57

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Several Search Methods Were Used

  • Depth First Search (DFS)
  • Depth-bounded Backtrack Search (DBS)
  • Iterative Broadening
  • Several rand values
  • We exploited search phase independence.

57

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Optimization for First Timetabling Instance

150 200 250 300 350 400 450 500 550 600 00:00 01:00 02:00 03:00 04:00 05:00 Solution Cost Time (minutes:secs) Iterative Broadening rand = 70.0 rand = 7.0 rand = 0.5 DFS DBS

58

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SLIDE 59

Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Results for the Second Instance

300 350 400 450 500 550 600 650 700 750 800 00:00 01:00 02:00 03:00 04:00 05:00 Solution Cost Time (minutes:secs) Iterative Broadening rand = 70.0 rand = 7.0 rand = 0.5 DFS DBS

  • Similar to the first instance

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Our Contributions: Conclusions

  • 1. Course Scheduling as a CSP
  • 2. Problem entities handled as resources
  • 3. An “armory” of search methods
  • 4. Design of semi-random heuristics
  • 5. Efficient lightweight constraint propagation schemas

60

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Future Perspectives

  • Automatically find best k threshold
  • Change search method during search
  • A mathematical semi-random heuristics framework

61

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Main References

  • R. Bart´

ak, “Incomplete depth-first search techniques: A short survey,” in CPDC2004: 6th Workshop on Constraint Programming for Decision and Control, Gliwice, Poland, 2004, pp. 7–14.

  • C. Bessi`

ere, “Constraint propagation,” in Handbook of Constraint

  • Programming. Amsterdam: Elsevier Science, 2006, ch. 3, pp.

29–83.

  • E. C. Freuder and R. Wallace, “Selective relaxation for constraint

satisfaction problems,” in ICTAI 1991: 3rd International Conference on Tools for Artificial Intelligence, San Jose CA, IEEE, 1991, pp. 332–339.

  • B. McCollum, A. Schaerf, B. Paechter, P. McMullan, R. Lewis,
  • A. J. Parkes, L. D. Gaspero, R. Qu, and E. K. Burke, “Setting the

research agenda in automated timetabling: The second international timetabling competition,” INFORMS Journal on Computing, vol. 22, no. 1, pp. 120–130, 2010.

  • M. L. Pinedo, Scheduling: Theory, Algorithms, and Systems,

4th ed. New York: Springer, 2012.

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Course Scheduling in an Adjustable Constraint Propagation Schema Nikolaos Pothitos, Panagiotis Stamatopoulos, Kyriakos Zervoudakis

  • 1. Introduction
  • 2. Related Work
  • 3. Contents
  • 4. Our Course

Timetabling Model

  • 5. New Random

Heuristics

  • 6. Adjusting

Constraint Propagation

  • 7. Conclusions –

Contributions

Thank You Very Much! I Will Be Happy to Answer Your Questions :-)

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