Filtering Algorithms for Global Constraints Classes of Constraints - - PowerPoint PPT Presentation

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints

Mats Carlsson

Swedish Institute of Computer Science (SICS)

November 12, 2005

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Constraint Programming Systems

◮ constraint network: domain variables and

constraints

◮ propagation loop ◮ search procedure

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

The CP View

Problem = conjunction of constraints

◮ constraint = subproblem ◮ trivial constraint: x < y ◮ complex constraint: all different([x1, . . . , xn]): the

xi variables must be pairwise different

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Problem Solving

Problem solving = propagation + search

◮ every constraint C comes with a filtering algorithm

(FA), aka. propagator

◮ the FA removes values that do not belong to any

solution of C from the domains of the variables

◮ the propagation loop applies FA’s until a fixpoint is

reached

◮ the search procedure adds tentative constraints at

such fixpoints

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Strength of Filtering

Notions of Consistency

◮ arc-consistency (AC): all values that do not belong

to any solution of C are removed (perfect filtering)

◮ sometimes called generalized arc-consistency

(GAC) in the context of non-binary constraints

◮ a weaker form, bound-consistency (BC), adjusts

the lower and upper bounds of domains

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Strength of Filtering

Example [Puget 1998]

Constraint Network all different([x, y, z, u, v, w]) x ∈ [3, 4], y = 7, z ∈ [2, 5], u ∈ [2, 7], v ∈ [1, 3], w ∈ [3, 4] Arc-Consistent Domains x ∈ [3, 4], y = 7, z ∈ {2, 5}, u ∈ {2, 5, 6}, v ∈ [1, 2], w ∈ [3, 4] Bound-Consistent Domains x ∈ [3, 4], y = 7, z ∈ [2, 5], u ∈ [2, 6], v ∈ [1, 2], w ∈ [3, 4]

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Global Constraints

Definition

C is a global constraint iff C is ...

◮ a non-binary constraint ◮ a conjunction of elementary constraints ◮ an abstraction for which we can have a FA with a

global view

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Global Constraints

Advantages

expressiveness they correspond to common structures in combinatorial problems reusability a library of global constraints is a library

• f reusable software components

efficiency they admit powerful filtering algorithms

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

An Application-Based Classification

◮ order constraints ◮ value constraints ◮ partitioning constraints ◮ timetabling constraints ◮ graph constraints ◮ scheduling constraints ◮ data constraints

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Order Constraints

min(x, [y1, . . . , yn]) max(x, [y1, . . . , yn]) min n(x, k, [y1, . . . , yn]) x is the kth smallest value max n(x, k, [y1, . . . , yn]) x is the kth largest value sorting([x1, . . . , xn], [y1, . . . , yn]) [y1, . . . , yn] is a permutation of [x1, . . . , xn] in nondecreasing order. lex chain([t1, . . . , tn]) [t1, . . . , tn] is a sequence of tuples in lexicographic order.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Value Constraints

among(k, [y1, . . . , yn], [v1, . . . , vm]) k vars in [y1, . . . , yn] take values in [v1, . . . , vm] common(k, l, [x1, . . . , xn], [y1, . . . , ym]) k vars in [x1, . . . , xn] take values in [y1, . . . , ym], and l vars in [y1, . . . , ym] take values in [x1, . . . , xn] same([x1, . . . , xn], [y1, . . . , yn]) [x1, . . . , xn] is a permutation of [y1, . . . , yn]

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Partitioning Constraints

all different([x1, . . . , xn]) ∀i, j : xi = xj symmetric all different([x1, . . . , xn]) ∀i, j : xi = xj ∧ (xi = j ≡ xj = i) nvalue(k, [x1, . . . , xn]) xi take k different values gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]) value vj occurs cj times in [x1, . . . , xn]

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Timetabling Constraints

change(k, [x1, . . . , xn], ◦) xi ◦ xi+1 holds k times inflexion(k, [x1, . . . , xn]) the sequence order switches from strictly increasing to strictly decreasing (or vice versa) k times

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Graph Constraints

Successor Notation [x1, . . . , xn] means from node i we go to node xi Predecessor Notation [x1, . . . , xn] means we came to node i from node xi cycle(k, [x1, . . . , xn]) there are k cycles in [x1, . . . , xn] inverse([x1, . . . , xn], [y1, . . . , yn]) [x1, . . . , xn] is the successor notation of a path, and [y1, . . . , yn] is its predecessor notation

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Scheduling Constraints

Task Notation task(s, d, e, r, m) means a task with start time s, duration d, end time e, resource consumption r, and machine assignment m cumulative([t1, . . . , tn], l) n tasks are scheduled on a single machine so that at any time, the total resource consumption does not exceed l cumulatives([t1, . . . , tn], [l1, . . . , lm]) n tasks are scheduled on m machines so that at any time, the total resource consumption of machine i does not exceed li

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Data Constraints

element(i, [v1, . . . , vm], x) vi = x table([t1, . . . , tm], [r1, . . . , rn]) each tuple ti is a member

• f the (ground) relation [r1, . . . , rn]

case(template, [t1, . . . , tm], dag) each tuple ti is a member of the (ground) relation encoded by dag

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

[Beldiceanu 2000a]

Global Constraints as Graph Properties of Structured Networks of Elementary Constraints of the Same Type

◮ Find out the basic constituents of the global

constraint.

◮ Classify the properties of each basic constituent. ◮ Understand how the properties interact.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

A Simple Global Constraint

Initial network

Arcs are associated with elementary constraints. Final network

Ask properties of sub-graph of elementary constraints that still hold.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

Graph Generators

LOOP

• SELF
• PATH

CHAIN

• CIRCUIT
• CYCLE
• PRODUCT
• SYMMETRIC PRODUCT
• GRID
• CLIQUE

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

Graph Characteristics

NVERTEX |V(G)| NEDGE |E(G)| NSOURCE number of vertices without predecessor NSINK number of vertices without successor NCC number of connected components of G MIN NCC number of vertices of smallest c.c. of G MAX NCC number of vertices of largest c.c. of G NSCC number of strongly connected components of G MIN NSCC number of vertices of smallest s.c.c. of G MAX NSCC number of vertices of largest s.c.c. of G

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

Graph Properties and Graph Invariants

◮ A graph property is a relation

C ◦ V, ◦ ∈ {≤, ≥, =, =}, where C is a graph characteristic and V is a domain variable.

◮ A graph invariant is a relation on graph

characteristics that is valid for a class of graphs. They form necessary conditions.

◮ Example:

MIN NSCC = MAX NSCC ⇒ NVERTEX ≥ MIN NSCC + MAX NSCC

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

The Global Constraint “Space”

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

Example: nvalue(NVAL, VARS)

arguments NVAL : dvar, VARS : collection(var − dvar) restrictions 0 ≤ NVAL ≤ |VARS| arc input variables arc generator clique arc constraint VARS.var[1] = VARS.var[2] graph properties NSCC = NVAL example nvalue(3, {var − 3, var − 1, var − 7, var − 1})

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

Example: nvalue(3, {var − 3, var − 1, var − 7, var − 1})

Initial network: variables unbound

• V1

=

• =
• =
• =
• V2

=

• =
• =
• V3

=

• =
• V4

=

• Final network: variables instantiated, NSCC = 3
• 3

=

• 1

=

• =
• 7

=

• 1

=

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

A Graph-Theoretic Classification

A Catalog of Global Constraints [BelCarRam05]

◮ 235 constraints described explicitly (by Prolog

facts).

◮ Reference pages and figures generated

automatically from Prolog facts.

◮ 157 graph invariants identified. ◮ About 30 arc generators. ◮ About 30 graph properties.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Adding Redundant Constraints

Constraint Network x = y, x = z1, . . . , x = zn y = z1, . . . , y = zn z1 = z2, . . . , z2 = zn · · · zn−1 = zn D(x) = D(y) = {0, 1} Propagation

• Nothing. But either x = 0, y = 1 or x = 1, y = 0, so no
• ther variables can take these values. So we can

remove them from the other variables.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Adding Redundant Constraints

Constraint Network all different([x, y, z1, . . . , zn]) D(x) = D(y) = {0, 1} Propagation 0 and 1 are removed from D(z1), . . . , D(zn).

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

GAC-Schema

GAC-Schema [Bessi` ere & R´ egin 1997]

◮ A generic framework for achieving AC for any

constraint.

◮ You just have to provide a way of computing a

solution.

◮ Works incrementally (notion of support).

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

GAC-Schema

Providing Solutions

◮ List of allowed tuples. ◮ List of forbidden tuples. ◮ Any algorithm, e.g. constraint checker. ◮ Solver re-entrance.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

GAC-Schema

Key Idea

tuple partial solution of the constraint support ground, valid tuple

◮ while the tuple is valid: do nothing ◮ otherwise, search for a new support

for the values it contains

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

GAC-Schema

Complexity

◮ For n variables, d values, S the cost of seeking a

support, AC is achieved in O(ndS).

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Finite Automata

[Beldiceanu, Carlsson & Petit 2004]

◮ Writing a constraint checker is much easier than

writing a filtering algorithm.

◮ A checker can often be expressed as a finite

automaton with counters.

◮ A finite automaton can be generalized to deal with

variable sequences, constraining the sequence to be recognized by the automaton.

◮ We have implemented a generic global

constraints, parameterized by a given finite automaton.

◮ Under certain conditions, the constraint achieves

AC.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Finite Automata

Example: inflexion(k, [x1, . . . , xn])

Definition Holds if k is the number of times that the sequence

• rder switches between strictly increasing and strictly

decreasing. Example inflexion(4, [3, 3, 1, 4, 5, 5, 6, 5, 5, 6, 3]) holds since we have the four subsequences 314, 565, 6556 and 563, which all contain such a switch.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Finite Automata

Automaton for inflexion

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep [Beldiceanu 2000b]

◮ Standard technique in the design of efficient

algorithms.

◮ Used in filtering algorithms [Beldiceanu &

Carlsson 2001].

◮ For pruning a conjunction of constraints with two

shared variables.

◮ For instance, the constraint that a rectange does

not overlap with a set of other rectangles.

◮ disjoint2([r1, . . . , rn]) expresses n such constraints.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep for Line Intersections

Iteration:

• 1. Move to the

next event point.

• 2. Update the

sweep line status.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep: Forbidden Regions

Definition The forbidden region of constraint C wrt. x and y are the two intervals [x, x] and [y, y] such that: ∀x ∈ [x, x], y ∈ [y, y] : C(x, y, . . .) is false Example The forbidden region of all different wrt. x and y is:

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep: Example

Task: adjust min(x) for a set

• f forbidden regions wrt. x

and y.

◮ Sweep line: x. ◮ Sweep line status: for

each y value, #forbidden regions containing (x, y).

◮ x = 0 is infeasible.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep: Example

Task: adjust min(x) for a set

• f forbidden regions wrt. x

and y.

◮ Sweep line: x. ◮ Sweep line status: for

each y value, #forbidden regions containing (x, y).

◮ x = 1 is infeasible.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Computational Geometry

Sweep: Example

x = 0, . . . , 3 are infeasible, but x = 4 is feasible.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]) [R´ egin 1994]

◮ A matching is a set of edges with distinct

endpoints.

◮ A matching M covers X if every vertex in X is an

endopint of M.

◮ A solution to all different(X) is a matching M

between X and its values such that M covers X.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn])

Berge’s Theorem An edge belongs to some maximum matching iff, for an arbitrary matching, it belongs to:

◮ either an even alternating path beginning

at a free vertex,

◮ or an even alternating cycle.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]): Example [R´ egin 1994]

Domains x1 ∈ {1, 2} x2 ∈ {2, 3} x3 ∈ {1, 3} x4 ∈ {3, 4} x5 ∈ {2, 4, 5, 6} x6 ∈ {5, 6, 7}

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]): Example [R´ egin 1994]

Matching

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]): Example [R´ egin 1994]

Alternating Paths and Cycles

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]): Example [R´ egin 1994]

Arc Consistent Domains x1 ∈ {1, 2} x2 ∈ {2, 3} x3 ∈ {1, 3} x4 ∈ {4} x5 ∈ {5, 6} x6 ∈ {5, 6, 7}

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

all different([x1, . . . , xn]): Complexity

◮ AC in O(n3/2d) from scratch. ◮ AC in O(knd) incrementally. ◮ BC in O(n + d) from scratch.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

An Auxiliary Constraint for Sudoku [Simonis 2005]

◮ Nine constraints: one for each value v ∈ [1, 9]. ◮ Each value v must occur exactly once in each row

and column.

◮ A matching between rows and columns. ◮ A special case of the colored matrix aka.

cardinality matrix constraint.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Matching Theory

An Auxiliary Constraint for Sudoku

• 1. Create row and column vertices.
• 2. Create edge (i, j) if v ∈ Mij.
• 3. Find a maximal matching.
• 4. Identify strongly connected components (SCCs)

in the reoriented graph.

• 5. Remove edges (i, j) connecting distinct SCCs.
• 6. Remove v from the corresponding Mij.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]) [R´ egin 1996]

◮ Value vj occurs cj times in [x1, . . . , xn]. ◮ Modelled as a flow from [v1, . . . , vm] to [x1, . . . , xn]. ◮ Arcs that cannot belong to a feasible flow

correspond to infeasible values.

◮ Infeasible arcs discovered in a way similar to

all different.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]): Example [R´ egin 1996]

Value Graph

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]): Example [R´ egin 1996]

A Feasible Flow

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]): Example [R´ egin 1996]

Feasible Arcs Only

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)]): Complexity

◮ AC in O(n3/2m) from scratch. ◮ BC in O(n + m) from scratch.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Flow Theory

An Auxiliary Constraint for Sudoku [Simonis 2005]

◮ Consider the conjunction of a column (row) and a

3x3 box.

◮ One sets of six variables must use the same

values as another set of six values.

◮ A special case of same with cardinality. ◮ AC is achieved by solving a flow problem.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Scheduling Theory

Common Notions

unary resource A non-sharable resource (machine). discrete resource A resource (machine) that is sharable up to a fixed capacity. task An activity to be placed in time, consuming some amount of resource on some machine.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Scheduling Theory

Filtering Algorithms for Unary Resources

Let Ω be a set of tasks and i, j ∈ Ω be single tasks. Edge-Finding Rules to infer that i must be scheduled before (after) all tasks in Ω, which allows to adjust the start (completion) time of i. Not-First/Not-Last Rules to infer that i cannot be scheduled before (after) all tasks in Ω, which allows to adjust the start (completion) time of i. Detectable Precedences Rules to infer that i must be scheduled before (after) j, which allows to adjust the start (completion) times of i and j. Can all be done in O(n log n) [Vil´ ım 2004].

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Pitfalls

Designing Constraints

◮ Allow parameters to be variables (not integers) for

better expressive power, e.g.: element(i, [v1, . . . , vm], x) cumulative([t1, . . . , tn], l) gcc([x1, . . . , xn], [(v1, c1), . . . , (vm, cm)])

◮ Group related parameters into collection of

• bjects, cf.:

cumulative( [t1, . . . , tn], l) cumulative( [s1, . . . , sn], [d1, . . . , dn], [e1, . . . , en], [r1, . . . , rn], l)

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Pitfalls

Complexity

◮ Complexity results assume no aliasing among

parameters.

◮ Complexity of what case:

◮ worst-case ◮ average-case

◮ Complexity of what operation:

◮ one run of the algorithm from scratch ◮ one run of the algorithm after some change ◮ reaching a leaf without backtracking ◮ exploring the whole tree

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Some Pitfalls

Complexity

The complexity of the algorithm depends on the cost

• f some services provided by the constraint kernel:

◮ adjusting domain bounds ◮ checking v ∈ D(x) ◮ removing v from D(x) ◮ given v ∈ D(x), finding the next value is

important, e.g. for SCC analysis These costs should be independent of the domain size.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Implementation Issues

Stateful Propagators

If a FA does not have state, each time it wakes up it has to:

◮ Parse and check the parameters. ◮ Allocate and deallocate memory. ◮ Create the necessary datastructures. ◮ Sacrifice any opportunity for being incremental.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Implementation Issues

Incrementality

◮ Examples of opportunities for being incremental:

◮ Maintain partitioning of parameters as (ground,

nonground).

◮ Arithmetic constraints can detect in O(1) time that

nothing can be pruned.

◮ all different reuses and augments the previous

graph and matching.

◮ gcc reuses and augments the previous graph and

flow.

◮ case does not revisit failed DAG nodes. ◮ disjoint2 maintains a witness for the sides of each

rectangle to avoid sweeps.

◮ ≤lex remembers where it exited the automaton.

◮ Problems with incrementality:

◮ Only valid while in the same branch. ◮ Normally, recompute from scratch upon

backtracking.

◮ If too much has changed, incrementality costs

more than recomputation.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Implementation Issues

Domination

◮ General idea: find a rule saying “if object o1 is

valid (cannot be pruned), then object o2 is also valid”.

◮ We say that o1 dominates o2. ◮ Example use: in disjoint2, for rectangles sharing

the same min(x), plus other conditions.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Implementation Issues

Fixpoints and Aliasing

Should the FA reach a fixpoint or not?

◮ Sometimes easy to achieve, sometimes not. ◮ Last resort: enclose the whole FA in a fixpoint

loop.

◮ Constraint kernel should accommodate fixpoint as

well as non-fixpoint FAs.

◮ Aliased parameters (possible in some host

languages) destroy fixpoint properties.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Methodology Issues

Performance Analysis

◮ Several complexity measures are relevant:

non-incremental one run of the algorithm from scratch incremental one run of the algorithm after some change amortized reaching a leaf without backtracking

◮ How to evaluate performance:

◮ worst-case complexity analysis ◮ benchmarking the FA in isolation ◮ benchmarking the FA in context of an application

◮ My favorite presentation method: scatter plots.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Methodology Issues

Debugging

◮ Writing a FA is tedious and error-prone. Common

errors:

• 1. false instances are accepted
• 2. too much pruning (loss of solutions)
• 3. too little pruning (loss of propagation)

◮ Debugging methodology: generate random

instances.

◮ Compare FA with an alternative, equivalent

implementation.

◮ Run FA once and compare domains. ◮ Collect all solutions and compare their numbers. ◮ Randomize domains as well as variable order and

value order.

◮ Study small instances to get inspiration about

more filtering rules.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

Outline

Constraint Programming Systems Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometry Matching Theory Flow Theory Scheduling Theory Final Remarks References

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

References

Jean-Charles R´ egin. A filtering algorithm for constraints of difference in CSP .

• Proc. AAAI-94, 1994.

Jean-Charles R´ egin. Generalized Arc Consistency for gcc Constraint.

• Proc. AAAI-96, 1996.

Christian Bessi` ere & Jean-Charles R´ egin. Arc Consistency for General Constraint Networks: Preliminary Results

• Proc. IJCAI-97, 1997.

Jean-Charles R´ egin. The Symmetric all different Constraint.

• Proc. IJCAI-99, 1999.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

References

Nicolas Beldiceanu. Global constraints as graph properties on structured network of elementary constaints of the same type.

• Proc. CP

, 2000. Nicolas Beldiceanu, Mats Carlsson, Jean-Xavier Rampon, Charlotte Truchet. Graph Invariants as Necessary Conditions for Global Constraints. SICS Technical Report T2005-07, 2005. Nicolas Beldiceanu, Mats Carlsson, Jean-Xavier Rampon. Global Constraint Catalog. SICS Technical Report T2005-08, 2005.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

References

Nicolas Beldiceanu, Mats Carlsson, Thierry Petit. Deriving filtering algorithms from constraint checkers.

• Proc. CP

, LNCS 3258, 2004. Helmut Simonis. Sudoku as a Constraint Problem.

• Proc. Workshop on Modelling and Reformulating

Constraint Satisfaction Problems at CP , 2005. Petr Vil´ ım. O(n log n) filtering algorithmsfor unary resource constraint.

• Proc. CP-AI-OR, 2004.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

References

Nicolas Beldiceanu. Sweep as a Generic Pruning Technique.

• Proc. TRICS Workshop at CP

, 2000. Nicolas Beldiceanu, Mats Carlsson. Sweep as a generic pruning technique applied to the non-overlapping rectangles constraint.

• Proc. CP

, 2001.

Filtering Algorithms for Global Constraints Mats Carlsson Constraint Programming Global Constraints Classes of Constraints Application-Based Graph-Based Filtering Algorithms Adding Redundant Constraints GAC-Schema Finite Automata Computational Geometr Matching Theory Flow Theory Scheduling Theory Final Remarks References

References

Jean-Franc ¸ois Puget. A fast algorithm for the bound consistency of all different constraints.

• Proc. AAAI-98, 1998.

kurt Mehlhorn & Sven Thiel. Faster Algorithms for Bound-Consistency of the sorting and the all different Constraints.

• Proc. CP

, 2000. Irit Katriel & Sven Thiel. Fast Bound Consistency for the gcc Constraint.

• Proc. CP

, 2003. Irit Katriel & Sven Thiel. Complete Bound Consistency for the gcc Constraint. Constraints 10(3), 2005.