Synthesis of Search Algorithms from High-level

CP Models

?

Samir A.Mohamed Elsayed

??

,Laurent Michel

Computer Science Department,University of Connecticut.

Abstract.The ability to specify CP programs in terms of a declara-

tive model and a search procedure is instrumental to the industrial CP

successes.Yet,writing search procedures is often dicult for novices or

people accustomed to model &run approaches.The viewpoint adopted in

this paper argues for the synthesis of a search fromthe declarative model

to exploit the problem instance structures.The intent is not to eliminate

the search.Instead,it is to have a default that performs adequately in the

majority of cases while retaining the ability to write full- edged proce-

dures.Empirical results demonstrate that the approach is viable,yielding

procedures approaching and sometimes rivaling hand-crafted searches.

1 Introduction

Constraint programming (CP) techniques are successfully used in various in-

dustries and quite successful when confronted with hard constraint satisfaction

problems.Parts of this success can be attributed to the considerable amount of

exibility that arises from the ability to write completely tailored search proce-

dures.The main drive is based on the belief that

CP = Model +Search

where the model provides a declarative specication of the constraints,while

the search species how to explore the search space.In some CP languages,the

search can be quite sophisticated.It can concisely specify variable and value se-

lection heuristics,search phases [14],restarting strategies [9],large neighborhood

search [1],exploration strategies like depth-rst-search,best-rst search,or lim-

ited discrepancy search [12] to name just a few.This exibility is mostly absent

in mathematical programming where the so-called black-box search is controlled

through a collection of parameters aecting pre-processing,cut generation,or

the selection of predened global heuristics.Users of mathematical programming

solely rely on modeling techniques and reformulations to indirectly in uence and

hopefully strengthen the search process eectiveness.

Newcomers discovering CP often overlook the true potential of open (i.e.,

white-box) search specication and fail to exploit it.The observation prompted

?

This work is partially supported through NSF award IIS-0642906.

??

The author is partially supported by Helwan University,Cairo,Egypt.

a number of eorts to rethink constraint programming tools and mold themafter

LP and MIP solvers by eliminating open search procedures in favors of intelligent

black-box procedures.Eorts of this type include [14] and [6] while others,e.g.,

[21] provide a number of predened common heuristics.Our contention is that it

is possible to get the best of both worlds:retaining the ability to write tailored

search procedures,and synthesizing instance-specic search procedures that are

competitive with procedures hand-crafted by experts.

The central contribution of this paper is Cp-as,a model-driven automatic

search procedure generator written in Comet [24].Cp-as analyzes a CP model

instance at runtime,examines the variable declarations,the arithmetic and log-

ical constraints,as well as the global constraints and synthesizes a procedure

that is likely to perform reasonably well on this instance.Empirical results on a

variety of representative problems (with non-trivial tailored search procedures)

demonstrate the eectiveness of the approach.The rest of the paper is organized

as follows:Section 2 presents related work.Section 3 provides details about the

synthesis process,while Section 4 illustrates the process on a popular CP appli-

cation.Experimental results are reported in Section 5 and Section 6 concludes.

2 Related Work

The oldest general purpose heuristics follow the fail-rst principle [11] and or-

der variables according to the current size of their domains.Impacts [18] were

introduced as a generic heuristic driven by the eect of labeling decisions on the

search space contraction.wdeg and dom/wdeg [3] are inspired by SAT solvers and

use con icts to drive a variable ordering heuristic.Activity-based search [15] is

driven by the number of variables involved in the propagation after each decision

and is the latest entry among black-box general purpose heuristics.

Minion [6] oers a black-box search and combines it with matrix based

modeling,aiming for raw speed alone to produce`model and run'solutions.

CPhydra [17] is a portfolio approach exploiting a knowledge base of solved in-

stances.It combines machine learning techniques with the partitioning of CPU-

time among portfolio members to maximize the expected number of solved in-

stances within a xed time budget.Model-driven derivation of search rst ap-

peared in [26] for Constraint-Based Local Search (CBLS).Given a model,a

CBLS synthesizer derives a local search algorithm for the chosen meta-heuristic.

It analyzes the instance and synthesizes neighborhoods as well as any other nec-

essary components.The Aeon synthesizer [16] targets the scheduling domain

where combinatorial structures are easier to recognize and classify.Note that

Aeon handles both complete and incomplete (CBLS) solvers.The rst exten-

sion to generic CP models was proposed in [4] and is extended here with a larger

rule set for global constraints that now uses many variable and value selection

heuristics.

3 The Synthesis Process

Cp-as denes rules meant to recognize combinatorial structures for which good

heuristics exist.Each rule,when red,produces a set of recommendations char-

acterized by a tness score,a subset of variables,and two heuristics for variable

and value selection as well as dynamic value symmetry breaking whenever ap-

propriate.This set of recommendations is the blueprint for the search itself.The

section describes the entire process.

3.1 Preliminaries

A CSP (Constraint Satisfaction Problem) is a triplet hX;D;Ci,where X is a set

of variables,D is a set of domains,and C is a set of constraints.Each x 2 X is

associated with a domain D(x),i.e.,with a totally ordered nite set (i.e.,a well-

ordered set) of discrete values over some universe U.A constraint c(x

1

; ;x

n

),

species a subset of the Cartesian product D(x

1

) D(x

n

) of mutually-

compatible variable assignments.X is the type of a variable,while X[] denotes

the type of an\array of variables".D = 2

U

is the type of a domain and C is the

type of a constraint.A COP (Constraint Optimization Problem) hX;D;C;Oi is

a CSP with an objective function O.

Common notations.vars(c) denotes the variables appearing in constraint c while

cstr(x) is the subset of constraints referring to variable x.The static degree

deg(x) of variable x is deg(x) =

P

c2cstr(x)

(jvars(c)j 1).Variables can be orga-

nized as arrays in the model and this is captured by a special tautological\con-

straint"array(x) that states that the subset of variables x X forms an array.

T(c) denotes the type of a constraint c (e.g.,knapsack,sequence,array,etc.).

Finally,T(C) = fT(c):c 2 Cg is the set of constraint types in C.

Denition 1.A variable ordering heuristic h

x

:X[]!N!X is a function

which,given an array of n variables [x

0

; ;x

n1

],denes a permutation of

0::n1 that produces a partial function [0 7!x

(0)

; ;n1 7!x

(n1)

]:N!X.

Example 1.The static variable ordering denoted h

static

simply produces the

variable ordering partial function h

static

= [0 7!x

0

; ;n 1 7!x

n1

].

Example 2.The static degree ordering denoted h

deg

uses a permutation :N!

N of 0::n 1 satisfying

8i;j 2 0::n 1:i j )deg(x

(i)

) deg(x

(j)

)

to dene the partial function h

deg

= [0 7!x

(0)

; ;n 1 7!x

(n1)

].

Example 3.The classic dom variable ordering denoted h

dom

will,when given an

array of variables x,uses a permutation :N!N of 0::n 1 satisfying

8i;j 2 0::n 1:i j )jD(x

(i)

)j jD(x

(j)

)j

to produce a partial function capturing a permutation of x.For instance,invoking

h

dom

([x

1

;x

2

;x

3

]) with D(x

1

) = f1;2;3g;D(x

2

) = f1g;D(x

3

) = f3;4g returns

the partial function [0 7!x

2

;1 7!x

3

;2 7!x

1

].The result produced by h

dom

is

dynamic,i.e.,the embedded permutation will use the domains of the variables

in x when it is invoked.

Example 4.The dom/wdeg [3] variable ordering denoted h

wdeg

will,when given

an array of variables x,use a permutation :N!N of 0::n 1 satisfying

8i;j 2 0::n 1:i j )

jD(x

(i)

)j

wdeg

(x

(i)

)

jD(x

(j)

)j

wdeg

(x

(j)

)

with

wdeg

(x

i

) =

P

c2C

weight[c]jvars(c) 3 x

i

^jfutV ars(c)j > 1.Following [3],

weight[c] is a counter associated to constraint c that tracks the number of con-

icts discovered by c during the search.The expression futV ars(c) denotes the

set of uninstantiated variables in c.

Denition 2.A value ordering heuristic h

v

:D!N!U is a function which,

given a domain d = fv

0

; ;v

k1

g of cardinality k,uses a permutation to

produce a serialization function for d dened as [0 7!v

(0)

; k 1 7!v

(k1)

].

Example 5.The min-value heuristic (denoted h

mv

) applied to the domain D(x) =

fv

0

; ;v

k1

g of a variable x uses a permutation :N!N satisfying

8a;b 2 0::k 1:a b )v

(a)

v

(b)

to produce a serialization partial function [0 7!v

(0)

; ;k 1 7!v

(k1)

].For

instance,invoking h

mv

(f3;7;1;5g) returns [0 7!1;1 7!3;2 7!5;3 7!7].

Denition 3.A value symmetry breaking heuristic h

s

:D!D is a function

that maps a set of k values from U to a subset of non-symmetric values.

3.2 Rules and Recommendations

Denition 4.Given a CSP hX;D;Ci,a rule r is a tuple hG;S;V;Hi where

G:2

C

!2

2

C

is a partitioning function that breaks C into G

1

G

n

such that

[

n

i=1

G

i

C and G

i

\G

j

=;8i 6= j.

S:h2

X

;2

D

;2

C

i!R is a scoring function,

V:h2

X

;2

D

;2

C

i!2

X

is a variable extraction function,

H:h2

X

;2

D

;2

C

i!hh

x

;h

v

i is a heuristic selection function.

All scores are normalized in 0::1 with 1 representing the strongest t.

Denition 5.Given a rule hG;S;V;Hi,a CSP hX;D;Ci,and a partition G(C) =

fG

1

G

n

g the rule's recommendations are fhS

i

;V

i

;H

i

i:0 < i ng with

S

i

= S(X;D;G

i

);V

i

= V(X;D;G

i

),and H

i

= H(X;D;G

i

).

Generic Partitioning Several rules use the same partitioning scheme

~

G.A rule r

focusing on constraints of type t uses the function

~

G to only retain constraints of

type t and yields one group per constraint.Namely,let n = jfc 2 C:T(c) = tgj

in

~

G(C) = ffc

1

g; ;fc

n

gg with all the c

i

constraints in C of type t.

3.3 Rules Library

Rules are meant to exploit combinatorial structures expressed with arrays,global

constraints,arithmetic constraints,and logical constraints.Structures can be

explicit (e.g.,global constraints),or implicit (e.g.,the static degree of a variable).

Cp-as oers one rule per combinatorial structure that can produce a set of

recommended labeling decisions.Global constraints play a prominent role in the

analysis and their rules are described rst.A brief discussion of a generic scoring

function used by most rules starts the section.

Generic Scoring The generic scoring applies to a group (i.e.,a subset) G C

of constraints and attempts to capture two characteristics:the homogeneity of

the entire set C and the coupling of the variables in each constraint of the group

G.A homogeneous constraint set contains few distinct constraint types that

might be easier to deal with.The homogeneity of C is measured by

1

jT(C)j

which

ranges in 0::1 and peaks at 1 when only one type of constraint is present in C.

The variable coupling for a single constraint c 2 G is an indicator of the amount

of ltering to be expected from c.

When vars(c) is a super-set of a user-specied array from the model,the

ratio of the maximal variable degree in vars(c) to the maximal overall degree

(r

1

(c) below) is used to estimate c's coupling.Otherwise,the simpler ratio r

2

(c)

is used.

r

1

(c) =

max

x2vars(c)

deg(x)

max

x2X

deg(x)

r

2

(c) =

jvars(c)j

j [

k2C:T(k)=T(c)

vars(k)j

The generic scoring function for G C is then

~

S(G) =

1

jT(C)j

max

c2G

r

1

(c) 9 a 2 C:T(a) = array ^vars(c) vars(a)

r

2

(c) otherwise

Observe how r

1

(c) and r

2

(c) are both in the range 0::1 delivering a generic

score in the 0::1 range.The rest of the section denes the rules.Each denition

species the partitioning G,scoring S,variable extraction V,and the heuristic

selection Hfunctions.Each of these function names is subscripted by a two letter

mnemonic that refers to the rule name.

Alldierent(ad) Rule.The alldifferent(x) constraint over the array x of n

variables holds when all variables are pairwise distinct.The rule uses the generic

partitioning

~

G and the generic scoring

~

S.The variable selection heuristic is simply

h

dom

(i.e.,the smallest domain),while the value selection heuristic is h

mv

(i.e.,

min-value).The variable extraction simply restricts the scope of the rule to

the variables of the constraint,namely V

ad

(X;D;fcg) = vars(c).The heuristic

selection H

ad

returns hh

dom

;h

mv

i.Note that V

ad

will always receive a singleton

as the rule uses the generic partitioning that always produces partitions with

singletons.The rule is thus h

~

G;

~

S;V

ad

;H

ad

i.

Knapsack(ks) Rule.The knapsack(w,x,b) constraint over the array x of n

variables holds when

P

n1

i=0

w

i

x

i

b.The knapsack rule uses the generic

partitioning

~

G and the generic scoring

~

S.

A customized variable ordering heuristic is desirable when a user-specied

array of variables coincides with the array x.If true,the rule favors a variable

ordering for x based on decreasing weights in w and breaks ties according to

domain sizes.Let :N!N be a permutation of the indices 0::n 1 into x

satisfying

8i;j 2 0::n 1:i j )hw

(i)

;jD(x

(i)

)ji hw

(j)

;jD(x

(j)

)ji

where denotes the lexicographic ordering over pairs.The variable ordering is

then a partial function h

ks

= [0 7!x

(0)

; ;n1 7!x

(n1)

].When x does not

correspond to a model array,the heuristic is simply h

dom

.The heuristic selection

function H is

H

ks

(X;D;fcg) =

h

ks

if 9 a 2 C:T(a) = array ^vars(a) = vars(c)

h

dom

otherwise

;h

mv

The variable extraction simply restricts the scope of the rule to the vari-

ables of the constraint,namely V

ks

(X;D;fcg) = vars(c).The rule is thus

h

~

G;

~

S;V

ks

;H

ks

i.

Spread(sp) Rule.The spread(x,s,) constraint over an array x of n variables

and a spread variable holds whenever s =

P

n1

i=0

x

i

^N

P

n1

i=0

(x

i

s=n)

2

holds.It constrains the mean to the constant s=n and states that is an upper

bound to the standard deviation of x [19].The rule uses the generic partitioning

and generic scoring functions.To minimize ,one must minimize each term in

the sum and thus bias the search towards values in D(x

i

) closest to s=n.This

suggests both a variable and a value selection heuristic.The value selection can

simply permute the values of the domain to rst consider those values closer to

s=n.Namely,let :N!N be a permutation of the range 0::k 1 satisfying

8i;j 2 0::k 1:i j )jv

(i)

s

n

j jv

(j)

s

n

j

in the denition of the value ordering h

vsp

= [0 7!v

(i)

; ;k 7!v

(k1)

] for

the domain D(x) = fv

0

; ;v

k1

g.Given h

vsp

,the ideal variable ordering is

maximum regret.Namely,the variable with the largest dierence between the

rst two values suggested by its h

vsp

ought to be labeled rst.Let :N!N be

a permutation for the range 0::n 1 satisfying

8i;j 2 0::n 1:i j )

h

vsp

(D(x

(i)

))(1) h

vsp

(D(x

(i)

))(0) h

vsp

(D(x

(j)

))(1) h

vsp

(D(x

(j)

))(0)

in the variable ordering h

xsp

= [0 7!x

(0)

; ;n 1 7!x

(n1)

].Note how the

value ordering h

vsp

:D!N!U is passed the domains of the two chosen vari-

ables x

(i)

and x

(j)

to form the regret between the best two values according to

h

vsp

.The heuristic selection H

sp

returns hh

xsp

;h

vsp

i and the variable extraction

V

sp

returns x (the variables of the spread) in the rule h

~

G;

~

S;V

sp

;H

sp

i.

Sequence(sq) Rule.The classic sequence(x,d,p,q,V) global constraint [2] re-

quires that for every window of length q in array x,at most p variables take

their values in V and the demands in d for values in V are met by the sequence.

The sequence rule overrides the partitioning function

~

G to group sequence con-

straints that pertain to the same sequence x and same demand d,to exploit the

tightness of the various sequencing requirement and to yield better variable and

value orderings.Let G

sq

(C) = fG

1

; ;G

k

g where G

1

through G

k

satisfy

8a;b 2 G

i

:vars(a) = vars(b) ^d(a) = d(b) ^T(a) = T(b) = sq

The rened scoring function

S

sq

(X;D;G) =

~

S(X;D;G)

P

c2G

U(c)

jGj

where U(c) =

c:q

c:p

P

j2c:V

d

j

n

scales the generic score

~

S with the average constraint tightness of a group G

and the tightness of a single sequence constraint.The tightness of sequence c is

proportional to c:q=c:p and to the overall demand for values in c:V.

Following [20],the ideal variable and value selection heuristics attempt to

avoid gaps in the sequence while labeling and give preference to values that

contribute the most to the constraint tightness.The permutation

x

:N!N of

0::n 1 satises

8i;j 2 0::n 1:i j!jx

x

(i)

n=2j jx

x

(j)

n=2j

(

x

prefers variables that are closer to the middle of the sequence) and is used to

dene the variable ordering h

xsq

= [0 7!x

x

(0)

; ;n1 7!x

x

(n1)

].The value

selection heuristic is driven by the tightness of a value j in all the constraints of

group G

U(j) =

X

c2G

U(c) (j 2 c:V )

The permutation

v

of the values in D(x) = 0::k 1 makes sure that i precedes

j in

v

if it has a higher utility,i.e.,

v

satises

8i;j 2 0::k 1:i j )

U(

v

(i))

U(

v

(j))

and leads to the value ordering h

vsq

= [0 7!v

v

(0)

; ;k 1 7!v

v

(k1)

].The

heuristic selection H

sq

returns hh

xsq

;h

vsq

i while the variable extraction function

V

sq

returns [

c2G

vars(c) for a group G of sequence constraints.The sequence

rule is hG

sq

;S

sq

;V

sq

;H

sq

i.

Weighted-Sum(ws) Rule.The rule applies to a COP hX;D;C;Oi with O

P

n1

i=0

w

i

x

i

where all the w

i

are positive coecients.The objective (without

loss of generality,a minimization) can be normalized as a linear constraint c

dened as o =

P

n1

i=0

w

i

x

i

with a fresh variable o.The partitioning function

G

ws

returns the singleton fo =

P

n1

i=0

w

i

x

i

g while the scoring function S

ws

always returns 1.To minimize the objective,it is natural to rst branch on

the term with the largest weight and choose a value that acts as the smallest

multiplier.Yet,variables in o are subject to constraints linking them to other

decision variables and it might be preferable to rst branch on those if these

variables are more tightly coupled.Let Z(x) = [

i20::n1

[

c2cstr(x

i

)

vars(c) nfxg

denotes the set of variables one\hop"away from variables in array x.The

decision to branch on x or on Z(x) can then be based upon an estimation of

the coupling among these variables.Like in the generic scoring,the expression

max

y2S

deg(y) can be used to estimate the coupling within set S and drives the

choice between x and Z(x) delivering a simple variable extraction function V

ws

V

ws

(X;D;fcg) =

x if max

y2x

deg(y) max

y2Z(x)

deg(y)

Z(x) otherwise

The variable ordering over x can directly use the weights in the objective.

But a variable ordering operating on Z(x) must rst determine the contributions

of a variable y 2 Z(x) to the terms of the objective function.Note how Z(y)\x

identies the terms of the objective function aected by a decision on y.It is

therefore possible to dene a weight function that aggregates the weights of the

term aected by a decision on y.Let

w(y) =

X

z2Z(y)\x

c:w(z):8y 2 Z(x)

denote the aggregate weights for variable y where c:w(z) is the actual weight

of variable z in the objective.A permutation :N!N of the variable indices

ranging over the n variables in x (respectively,over the n variables in Z(x))

satises

8i;j 2 0::n 1:i j )w(x

(i)

) w(x

(j)

)

and is key to dene the variable ordering h

ws

= [0 7!x

(0)

; ;n1 7!x

(n1)

].

The heuristic selection function H

ws

returns hh

ws

;h

mv

i (the value selection is

min-value) and the entire rule is hG

ws

;S

ws

;V

ws

;H

ws

i.

Pick-Value-First(pv) Rule.If the number of values to consider far outnumbers

the variables to label,it is desirable to rst choose a value and then a variable

to assign it to.This rule generates one recommendation for each variable array

and the partitioning function is thus G

pv

(C) = ffcg 2 C:T(c) = arrayg.The

scoring function measures the ratio array size to number of values

S

pv

(X;D;farray(x)g) =

(

1

jxj

j[

a2x

D(a)j

if j [

a2x

D(a)j jxj

0 otherwise

The variable extraction function V

pv

simply returns the variables in the array x

while H

pv

(X;D;farray(x)g) = hh

static

;h

mv

i.The rule is hG

pv

;S

pv

;V

pv

;H

pv

i.

Degree(deg) Rule.The rule partitions C with G

deg

(C) = ffcg 2 C:T(c) =

arrayg and issues and uses a scoring that conveys the diversity of the static

degrees of the variables in the arrays.The index of diversity is based on the

relative frequencies of each member of the collection [8] and is the rst factor

in the denition of S

deg

.The index tends to 1 for diverse populations and to 0

for uniform populations.The second factor captures the relative coupling of the

variables in the array and also belongs to the 0::1 range.The score function is

S

deg

(X;D;farray(x)g) =

1

z

X

d=1

p

2

d

!

max

y2x

deg(y)

max

y2X

deg(y)

where z is the number of distinct degrees,p

d

= freq

d

=jxj and freq

d

= jfa 2

x:deg(a) = dgj.Note that,when all the variables in x have the same static

degree the diversity index is equal to 0,sending the overall score to 0.The

variable extraction is V

deg

(X;D;farray(x)g) = x.The variable ordering follows

h

deg

,i.e.,it selects variables with largest degree rst.The value selection is h

mv

leading to a denition for the heuristic selection H

deg

that returns hh

deg

;h

mv

i

and the rule is hG

deg

;S

deg

;V

deg

;H

deg

i.

The Default Rule.The rule ensures that all variables are ultimately labeled and

its score is the lowest (i.e.,a small constant bounded away from 0).The rule

could eectively use any black-box heuristic like Activity-based search,Impact-

based search,dom=wdeg,dom=ddeg,or even the simple dom heuristic.In the

following,it defaults to the dom heuristic.G

def

(C) = C,S

def

(X;D;C) = .

V

def

(X;D;C) = X to make sure that all variables are labeled.The variable or-

dering is h

dom

and the value ordering is h

mv

.The overall heuristic selection func-

tion H

def

returns hh

dom

;h

mv

i and the rule boils down to hG

def

;S

def

;V

def

;H

def

i.

3.4 Symmetry Breaking

The symmetry-breaking analysis is global,i.e.,it considers the model as a whole

to determine whether symmetries can be broken dynamically via the search

procedure.When conclusive,the analysis oers a partitioning of the values into

equivalence classes that the search can leverage.

While breaking symmetries statically is appealing for its simplicity,it can

interfere with the dynamic variable and value selection heuristics.Breaking sym-

metries dynamically through the search sidesteps the issue.A global symmetry

analysis of the model identies equivalence classes among values in domains and

avoid the exploration of symmetric labeling decisions.The automatic derivation

of value symmetry breaking in Cp-as follows [23,5],where the authors propose

a compositional approach that detects symmetries by exploiting the properties

of the combinatorial sub-structures expressed by global constraints.

1 forall(r in rec.getKeys()) by (recfrg.getScore()) f

2 recfrg.label();

3 if (solver.isBound()) break;

4 g

Fig.1.A Skeleton for a Synthesized Search Template.

3.5 Obtaining and Composing Recommendations

Given a CSP hX;D;Ci and a set of rules R,the synthesis process computes a

set of recommendations rec dened as follows

let fG

1

; ;G

k

g = G

r

(C)

in

rec =

S

r2R

([

i21::k

fhS

r

(hX;D;G

i

i);V

r

(hX;D;G

i

i);H

r

(hX;D;G

i

i)ig)

Namely,each rule decomposes the set of constraints according to its parti-

tioning scheme and proceeds with the production of a set of recommendations,

one per partition.When a rule does not apply,it simply produces an empty set

of recommendations.Once the set rec is produced,the search ranks the recom-

mendation based on their scores and proceeds with the skeleton shown in Figure

1.Line 2 invokes the polymorphic labeling method of the recommendation.The

search ends as soon as all the variables are bound (line 3).Note that since

[

hS

r

;V

r

;H

r

i2rec

([

x2V

r

) = X

the search is guaranteed to label all the variables.Figure 2 depicts the label

method for a variable rst recommendation,i.e.,a recommendation that rst

selects a variable and then chooses a value.Line 10 retrieves the variables the

recommendation operates on,and line 12 selects a variable according to the

variable ordering h

x

embedded in the recommendation.Line 14 retrieves the

values that are to be considered for the chosen variable pxi.The getValues

method is responsible for only returning non-symmetrical values when value

symmetries can be broken (it returns the full domain of pxi otherwise).The

index vr spans over the ranks of these values in d and line 17 retrieves the vr

th

value from d.If the value is still in the domain,line 19 uses it to label pxi.

Line 24 alludes to the fact that value-rst recommendation also have their own

implementation of the Recommendation interface to support their control ow.

4 A Walk-through Example

The synthesis process is illustrated in detail on one representative COP featuring

arithmetic,reied as well as global constraints.In the scene allocation problem,

shown in Figure 3,one must schedule a movie shoot and minimize the production

costs.At most 5 scenes can be shot each day and actors are compensated per

day of presence on the set.The decision variable shoot[s] (line 2) represents

the day scene s is shot while variable nbd[a] represents the number of days an

actor a appears in the scenes.

1 interface Recommendation f

2 void label();

3 var<CP>fintg[] getVars();

4 setfintg getValues(var<CP>fintg x);

5 int hx(var<CP>fintg[] x,int rank);

6 int hv(setfintg vals,int rank);

7 g

8 class VariableRecommendation implements Recommendation f...

9 void label() f

10 var<CP>fintg[] x = getVars();

11 forall(rank in x.getRange()) f

12 var<CP>fintg pxi = hx(x;rank);

13 if (jD(pxi)j == 1) continue;

14 setfintg d = getValues(pxi);

15 int vr = 0;

16 while (vr < jdj) f

17 int pvr = hv(d,vr++);

18 if (pvr 2 pxi)

19 try<cp> cp.label(pxi;pvr);j cp.di(pxi;pvr);

20 g

21 g

22 g

23 g

24 class ValueRecommendation implements Recommendation...

Fig.2.The Variable/Value Recommendation Classes.

The objective function is a weighted sum leading to a score of 1 for the ws

rule.On a given instance,all the nbd variables have the same static degree (16)

while the remaining variables (shoot) all have a static degree of 18.Therefore,

max

y2nbd

deg(y) < max

y2shoot

deg(y) and the rule recommends to branch on the

connected (1-hop away) variables Z(nbd),i.e.,on the shoot variables.The rule

proceeds and creates synthetic weights for each entry in shoot that aggregates

the weight of terms in uenced by the scene being shot.

Beyond ws,two rules produce additional recommendations.The degree rule

produces a single recommendation,while the default rule produces another.Yet,

the score of the degree rule is 0 since all the variables have the same degree forcing

the diversity index to 0.The default rule issues a recommendation with a score

of to label any remaining variables not handled by the ws recommendation.

The value-symmetry analysis determines that the value (days) assigned to the

scenes (i.e.,shoot) are fully interchangeable as reported in [13].The symmetries

are broken dynamically with the getValues method of the recommendation.

The method returns the subset of values (days) already in use (these are no

longer symmetric and each one forms one equivalence class) along with one

unused day.Comparatively,the tailored search in [22] iterates over the scenes

1 Solver<CP> m();

2 var<CP>fintg shoot[Scenes](m,Days);

3 var<CP>fintg nbd[Actor](m,Days);

4 int up[i in Days] = 5;

5 minimize<m> sum(a in Actor) fee[a] nbd[a] subject to f

6 forall(a in Actor)

7 m.post(nbd[a]==sum(d in Days) (or(s in which[a]) shoot[s]==d));

8 m.post(atmost(up,shoot),onDomains);

9 g

Fig.3.A Model for the Scene Allocation Problem.

and always chooses to rst label the scene with the smallest domain and to break

ties based on the costliest scene rst.

5 Experimental Results

Experiments were carried out on a mix of feasible CSP and COP that benet

fromnon-trivial tailored search procedures.Each benchmark is executed 25 times

with a timeout at 300 seconds.Results are reported for Activity Based Search

(ABS),Impact-Based Search (IBS),Weighted Degree search (WDeg),a state-

of-the-art hand-written tailored search,and the search synthesized by Cp-as.

ABS,IBS and WDeg all use a slow restarting strategy based on an initial

failure limit of 3 jXj and a growth rate of 2 (i.e.,the failure limit in round i is

l

i

= 2l

i1

.Table 1 reports the average CPUtime (T) (in seconds),its standard

deviation (T) and the number of runs that timed out (TO).The analysis time

for Cp-as is negligible.Timeouts are\charged"300 seconds in the averages.

The tailored search procedures are taken from the literature and do ex-

ploit symmetry breaking when appropriate.The steel mill instances come from

CSPLib [7,10] and the hand-crafted search is from [25].The car sequencing in-

stances come from CSPLib and the tailored search uses the best value and vari-

able orderings from [20].The nurse rostering search and instances are from [19].

The progressive party instances come from[7] and the tailored search labels a pe-

riod fully (using rst-fail) before moving to the next period.The multi-knapsack

as well as the magic square instances are from [18].The tailored search for the

magic square uses restarts,a semi-greedy variant of h

dom

for its variable order-

ing and a randomized (lower or upper rst) bisection for domain splitting.Grid

coloring and radiation models and instances were obtained from the MiniZinc

Challenge

1

.All the COP searches are required to nd a global optimum and

prove optimality.All results are based on Comet 3.0 on 2.8 GHz Intel Core 2

Duo machine with 2GB RAM running Mac OS X 10.6.7.

Rule's adequacy The intent of Cp-as was to produce code reasonably close to

procedures produced by experts and competitive with generic black-box searches.

The evaluation suite contains additional benchmarks (quite a few classic CSP)

1

Available at http://www.g12.csse.unimelb.edu.au/minizinc/

Benchmark

Tailored

CPAS

ABS

IBS

WDEG

(T)

(T)

TO

(T)

(T)

TO

(T)

(T)

TO

(T)

(T)

TO

(T)

(T)

TO

car-1

0.1

0.0

0

0.1

0.0

0

80.7

63.1

1

300.0

0.0

25

88.3

111.1

5

car-2

0.1

0.0

0

0.1

0.0

0

38.8

42.2

0

221.2

95.7

14

53.7

78.9

1

car-3

0.7

0.1

0

0.6

0.0

0

266.4

66.1

19

300.0

0.0

25

276.8

80.2

23

debruijn

0.6

0.1

0

0.5

0.0

0

300.0

0.0

25

301.2

0.7

25

300.0

0.0

25

gap

13.8

1.3

0

10.5

0.3

0

44.7

2.3

0

15.4

0.6

0

91.1

5.9

0

golomb

3.4

0.3

0

2.6

0.2

0

32.3

14.8

0

137.2

60.0

1

15.3

0.2

0

color

24.9

2.5

0

193.6

0.7

0

300.0

0.0

25

300.0

0.0

25

300.0

0.0

25

gcolor(5-6)

2.7

0.2

0

2.3

0.1

0

22.1

14.3

0

5.3

0.8

0

83.8

1.0

0

knapCOP-1

0.8

0.0

0

0.0

0.0

0

0.5

0.0

0

0.4

0.1

0

1.3

0.0

0

knapCOP-2

10.4

0.1

0

3.2

0.1

0

2.7

0.5

0

5.0

2.0

0

13.4

0.3

0

knapCOP-3

300.0

0.0

25

34.6

0.5

0

64.7

13.8

0

213.7

64.4

3

300.0

0.0

25

knapCSP-1

0.6

0.0

0

0.1

0.0

0

0.1

0.1

0

0.1

0.1

0

0.3

0.2

0

knapCSP-2

3.2

0.0

0

0.8

0.0

0

1.0

0.4

0

2.0

1.0

0

4.2

2.5

0

knapCSP-3

300.0

0.0

25

9.3

0.1

0

13.8

11.4

0

63.0

37.9

0

282.4

40.8

20

magic Sq-10

4.6

4.2

0

300.0

0.0

25

2.3

2.8

0

1.3

1.6

0

89.2

126.5

6

magic Sq-11

7.9

9.2

0

300.0

0.0

25

9.2

17.2

0

3.8

2.5

0

249.6

87.6

16

magicseries

5.8

0.7

0

5.7

0.1

0

2.8

1.6

0

2.7

0.4

0

1.9

3.0

0

market

5.8

0.2

0

5.2

0.1

0

30.4

21.9

0

37.2

27.1

0

47.3

36.5

0

nurse(z3)

0.3

0.0

0

4.6

0.1

0

40.3

17.2

0

18.9

6.9

0

163.4

49.8

0

nurse(z5)

0.1

0.0

0

2.1

0.0

0

53.8

6.0

0

13.0

9.4

0

61.4

15.6

0

perfectSq

0.2

0.0

0

0.2

0.0

0

300.0

0.0

25

300.0

0.0

25

300.0

0.0

25

progressive1

0.1

0.0

0

0.1

0.0

0

67.3

96.4

2

41.1

34.7

0

3.7

2.3

0

progressive2

0.6

0.0

0

0.7

0.0

0

112.3

125.7

7

278.8

64.6

22

175.4

114.8

9

progressive3

0.1

0.0

0

0.1

0.0

0

19.2

58.8

1

46.2

84.4

2

153.4

142.7

11

radiation1

2.3

0.0

0

0.5

0.0

0

0.4

0.1

0

1.7

0.1

0

0.1

0.0

0

radiation2

7.3

0.8

0

300.0

0.0

25

2.2

0.4

0

8.7

1.3

0

0.6

0.2

0

radiation3

198.1

3.9

0

300.0

0.0

25

0.5

0.1

0

2.4

0.2

0

0.1

0.0

0

radiation4

2.0

0.0

0

0.2

0.0

0

1.0

0.2

0

5.5

0.2

0

0.6

0.3

0

radiation5

0.0

0.0

0

12.4

0.1

0

1.1

0.2

0

5.6

0.5

0

0.3

0.2

0

radiation6

1.1

0.0

0

300.0

0.0

25

1.1

0.2

0

6.5

0.9

0

0.2

0.0

0

radiation7

1.3

0.0

0

11.2

0.3

0

1.3

0.4

0

10.0

0.7

0

0.2

0.1

0

radiation8

6.8

0.0

0

300.0

0.0

25

2.1

0.6

0

9.5

2.8

0

0.5

0.1

0

radiation9

300.0

0.0

25

4.8

0.1

0

2.1

0.5

0

9.5

0.7

0

1.1

0.6

0

RRT

4.7

0.0

0

4.9

0.1

0

145.4

131.9

10

243.2

105.9

17

91.3

94.8

3

scene

0.4

0.0

0

0.7

0.0

0

156.7

45.8

0

47.1

16.1

0

300.0

0.0

25

slab1

5.3

0.0

0

2.6

0.4

0

300.0

0.0

25

300.0

0.0

25

290.6

47.2

24

slab2

2.9

0.0

0

3.6

0.2

0

300.0

0.0

25

300.0

0.0

25

266.3

93.5

22

slab3

300.0

0.0

25

7.7

0.5

0

300.0

0.0

25

300.0

0.0

25

288.1

59.5

24

sport

6.6

1.1

0

5.4

0.1

0

151.2

123.9

7

255.3

96.3

20

131.8

111.1

5

Total

1526

100

2131

150

3170

197

4113

279

4428

294

Table 1.Experimental Results.

that terminate extremely quickly for all the search algorithms and are therefore

providing no insights into Cp-as's behavior.Figure 4 graphically illustrates how

often the various rules contribute to the search procedure of a model.Unsurpris-

ingly,a rule like\pick-value-rst"is used extremely rarely (only on the perfect

square) as the overwhelming majority of benchmarks do not have this property.

The other rules are used substantially more often.The fallback rule is used fairly

rarely as well.Overall,the rules do not overt the benchmarks,i.e.,we are far

from equating one-rule with one benchmark.

Tailored Search Procedures written by experts are often sophisticated with sym-

metry breaking and rich variable/value ordering using multiple criteria.The per-

formance of customsearches is therefore a target to approach and possibly match

on a number of benchmarks.Cp-as is successful in that respect and only falls

short on models like radiation (it cannot generate a bisecting search),or graph

coloring (it branches on the chromatic number too early).On the magic square

Cp-as cannot exploit semantics not associated with any one global constraint.

Fig.4.Rule Usage.

Black-box searches Compared to

black-box searches,Cp-as is gener-

ally competitive,especially in terms

of robustness.Sometimes,the black-

box heuristics performbetter (e.g.,on

radiation) and this needs to be fur-

ther investigated with a much length-

ier set of experiments with and with-

out restarting.Finally,it is possible

and maybe even desirable to switch

to a fallback rule that uses an eec-

tive black-box search techniques that

dominates a plain domheuristic.This

was intentionally left out to avoid con-

fusions about the true causes of Cp-

as behavior.The grand total of running times and number of timeouts across

the 5 searches is particularly revealing.

6 Conclusion

Cp-as automatically generates search algorithms from high-level CP models.

Given a CP model,Cp-as recognizes and classies its structures to synthesize

an appropriate algorithm.Empirical results indicate that the technique can be

competitive with state-of-the-art procedures on several classic benchmarks.Cp-

as is able to generate searches that split variables into groups/phases and uses

specialized variable and value ordering heuristics within each group.Cp-as also

relies on a global value symmetry breaking analysis that follows [23,5] and whose

results are exploited within each group of variables.

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