A LGPL framework for combinatorial optimization &

problem solving

Renaud De Landtsheer

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Content

• What are combinatorial problems?• Several approaches for solving combinatorial

problems• What is Asteroid• Roadmap

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Combinatorial problem solving

• Given a problem: • A set of unknown, with discrete range

• hence, combinatorial problem• A set of constraints involving the unknowns• (An objective function to minimize / maximize)

• Find a value for each variable, such that• The values belong to the ranges of the

variables• Each constraint is enforced• (The objective is minimized / maximized)

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Examples of combinatorial problems

• Scheduling problems eg: Job shop: • Set of tasks eg: 1k• With precedence constraints and ressource

usage• Minimize makespan

• Decision problems eg: Nqueens: • Given a chessboard of size N*N (eg: 1000)• Put N queens on it such that they do not

threaten each other• Routing problems eg: tsp+deadline

• Visit a set of places, with deadline constraints• Minimize overall distance

• Etc.

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Approaches for solving combinatorial problems

• Exhaustive symbolic exploration • Constraint programming• MIP

• Local search -based• Constraint-based local search• Large neightboorhood search

• Other approaches• Column generation• Ant colony

Method should be tailored to requirements!

COMET, Kangaroo

(COMET, Scampi)

All CP engines

LocalSolver

GLPK, LPSolve…

Gecode, Jacop, CHOCO, COMET, Scampi

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Relevant aspects to be considered

• Scalability• CBLS provides very good scalability, better

than CP• Optimality

• CBLS might be suboptimal; requires benchmarking

• Development cost• The more declarative is the framework, the

lower• Need for tuning

• MIP requires zero (but high modeling cost)• CP requires more, CBLS more than CP

• Specific requirements• Local search requires adequate search

heuristics

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What bring these framework compared to a specific

development?• Declarativity

• Code your problem as it is defined, through standardized vocabulary, eg: constraints• allDiff(Array_of_variables)• a <= b+5

• Lower development cost• Thanks to declarativity, focus given to the fine

tuning, not to heavy development• Efficiency

• Standardized constraints are deeply optimized• Eg: Alldiff costs O(1) for each move in Asteroid

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Local search

• Perform a descend in the solution space; repeatedly move from one solution to a better one

• Next solution identified via neighborhood exploration• Neighborhood = set of neighbors obtained by slightly

perturbing the solution (several possibilities)• Nqueens: moving a queen or swapping two queens• Scheduling: swapping two tasks• Routing: moving a task

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Constraint-based local search

• Enables very fast evaluation of neightbors• Fastest than evaluating objective from scratch

• Additional mechanisms are provided to identify the most relevant neightbors without evaluating them• Eg: get the worse part of the problem

• Rich library of standardized formulas that can be used to encode a problem into the Asteroid framework

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A standard Benchmark of Asteroid

1k 2k 3k 4k 5k 6k 7k 8k 9k 10k0

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30

40

50

60

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Run time of NQueen(swap) wrt number of queens

+ N² fitting

TtotalPolynomial (Ttotal)

run

tim

e [

secon

ds]

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Logical Architecture of Asteroid

PropagationPropagation element, propagation structure

ComputationVariable, invariant, IntVar, IntSetVar

Library of invariantsPlus, minus, argMax, Inter, ArrayAccess, …

Constraint structureConstraint system, constraint

Constraint libraryeq, neq, ge, alldiff, atMost

SearchselectMax,…

stopwatch

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Availability and Roadmap of Asteroid

• Asteroid is a research prototype for Wist3.0 project (Wallonie) • Developed entirely at CETIC• Aiming at prioritizing patients in oncology

services• Includes specific extensions for scheduling

problems

• Available free of charge under LGPL license• Bug correction, and some extensions are LGPL

• Written entirely in Scala

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NQueens naïve: the initial blabla

object NQueens extends SearchEngine{ def main(args: Array[String]) {

val N:Int=40 val range:Range = Range(0,N) val MaxIT = 10000

println("NQueens(" + N + ")")

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NQueens naïve : defining the problem

val m: Model = new Model val Queens:Array[IntVar] = new Array[IntVar](N) for (q <- range){Queens(q) = new IntVar(m, 0, N-1, q, "queen" + q)}

val c:ConstraintSystem = new ConstraintSystem(m)

c.post(AllDiff(for ( q <- range) yield (q plus Queens(q)).toIntVar)) c.post(AllDiff(for ( q <- range) yield (q minus Queens(q)).toIntVar))

c.close m.close

Plus and minus are infix expression for sum and minus invariants

AllDiff constructor does not require « new »(scala case class)

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NQueens naïve : defining the search procedure

var it:Int =0

while((c.Violation.getValue() > 0) && (it < MaxIT)){

val (q1,q2):(Int,Int) = selectMin2(range, range, (q1:Int, q2:Int) => c.getSwapViol(Queens(q1), Queens(q2)), (q1:Int,q2:Int) => q1 < q2)

Queens(q1) :=: Queens(q2)

it += 1 println("it: " + it + " " + c.Violation + " (swapped "+ q1 + " and " + q2 + ")") }

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NQueens naïve : printing & running

NQueens(40)it: 1 Violation:=38 (swapped 12 and 9)it: 2 Violation:=37 (swapped 8 and 39)…it: 33 Violation:=1 (swapped 10 and 14)it: 34 Violation:=0 (swapped 3 and 11)Violation:=0IntVars(queen0:=13,queen1:=17,queen2:=25,queen3:=11,queen4:=8,…

println(c.Violation) println(m.getSolution(true))}

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Invariant library

• Logic• Acces on array of IntVar,

IntSetVar• Access on array of IntVar,

where index is IntSetVar• Sort• Filter, Cluster (indexes of

element whose value is…)• SelectLeSetQueue (for Tabu

search)• …

• MinMax• Min, Max (linear and Log

variants)• ArgMin, ArgMax

• Numeric• sum, prod, minus, div• Also on enumeration of

IntVar• abs

• Set• Cardinality• Inter, Union, Diff• Interval, MakeSet

• Look also in Computation Structure• Identity invariants• Events « TriggerOn »

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Constraint Library

• Global• allDiff• atLeast• atMost (untested)

• Basic• Eq, Neq• G, GE• L, LE

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Need more info

• Asteroid Website• Wiki

https://forge.pallavi.be/projects/asteroid/wiki• Svn https://svn.forge.pallavi.be/asteroid

• Scaladoc available• Also on the website

• Renaud De Landtsheer• rdl@cetic.be

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References

Constraint-Based Local Searchby Pascal van Van Hentenryck, Laurent Michel