jf - NASA Technical Reports Server (NTRS)

3 downloads 0 Views 2MB Size Report
Ames. Research. Center. Artificial Intelligence Research Branch. Technical ... B. Philips. _. Mark. D. Johnston. 2. Philip. Laird 3. 1Sterling Federal Systems ...... (1+ (eli. *num-col-elts* col). )) (setf. (elt. _num-dn-diag-elts*. (+ limit. (- row col))). (1+.

?',j f_

The

Min-Conflicts

Experimental

and

Heuristic: Theoretical

STEVE

MINTON

ANDREW MARK

B. D.

RESEARCH NASA

LAIRD

BI_ANCH, AMES

MOFFETT

(_ASA)

MAIL

RESEARCH FIELD,

(F_ASA- T_-IO7877) THE HEURISTIC: EXPERIMENTAL Rr SULTS

27

PHILIPS

JOHNSTON

PHILIP AI

Results

CA

STOP

244-17

CENTEI_ 94035

N_2-25445

MIN-CONFLICTS AND THEORETICAL

p Unclas O091blO

G]/6]

|

_A._A

Ames Artificial

Research Intelligence Technical

Center Research Branch Report

FIA-91-25

September

1991

REPORT P!J_liC

re_ft_n(_

burden

_cr

this

collection

gathenng and ma_ntaimng the data ¢oltection of information, including _av_5 H.'qh_ay, Suite )204. Arlington,

DOCUMENTATION of

bnformatron

is est,mated

needed, and completing sugcjestions for reducing VA 22202 J302. _nd

1. AGENCY USE ONLY (Leave blank)

and this to the

to

PAGE

average

t hour

reviewing the collection burden, to Washin 100 the

we found

hill-climbing

by our

phase,

in the

For

our

fewest

Interestingly,

network.

required

the

a repair,

column

ties randomly).

that

with

To make

different

with

as follows.

methods are

for

compared

Consequently,

only

intialization.

I I "= I°'I" = I° l" = I° l" = i° l" = io ] I co.mcts ter i ti zatio. 1 3.n 1 7.35 1 9.75 1 0.90 l 12.02I 2.S0 l Table In the first the

number

method

1: Number each

of queens to the

geometric

min-conflicts

heuristic

then

90% of the linear

variables

number

queen

involved

be attributed

of Conflicts is placed

in conflicts properties

took

in conflict,

about the

for N-Queens

in a randomly was of the

found

works

of steps.

6

chosen

column.

Experimentally,

to be approximately

problem.

.6N steps

heuristic

Algorithms

to find

Repair a solution.

extremely

well;

strategies Note a solution

.9N.

This

employing that

even

is found

can the with in a

In the secondinitialization method the system puts one queen in each column, with preferencegiven to columnsthat haveno conflicts. To accomplishthis, the systemmaintains a list of empty columns. When placing a queenin a given row, the system examinesthe list of empty columns, looking for a position with no conflicts (i.e., a position with no conflicts along the diagonals). If more than one column is found, the systemselectsamong them randomly. If none are found, the systemrandomly selectsa position from the list of empty columns. This method performs significantly better than the first initialization method becausethe number of conflicts producedgrows very slowly. However, it doesa bit worse than the initialization method usedin the experiments describedabove,which involves a slight variation. In this third method, an initial assignmentis created using a greedy algorithm that iterates through the rows,placing eachqueenon the column whereit conflicts with the fewestpreviously placed queens(breakingties randomly). Table 1 shows number of conflicts as N becomeslarge. Strategy BasicBacktrackf Most ConstrainedBacktrackJ MinConflictsHill-Climbing$ MinConflictsBacktrack_ i = number

entry

Each

in the

table

table

with entry

fully. the

The

informed row

the

the

22150 (81%)

*

*

*

57.0

55.6

48.8

48.5

52.8

48.3

46.8

25.0

30.7

27.5

27.8

26.4

row

was

number on the

experiments

find

shows

completely using

program to place

that

movements. the

for

the

a queen.

the

behavior.

three-quarters

At n = 1000,

row that

of these

successful

program

runs

row first"

study

n =

10 6

required

table

heuristic. constrained

of the

7

than

a

of time;

terminated

and

the corresponding For

shows

the

program

when

choosing problem,

for the program

on only

is either

completed

This

n-queens

Each

movements

amount

program.

in the

a solution

fewer

backtracking million.

move

it was

the program

powerful heuristic Reingold[3]. The found

each

this occurred,

backtracking

is most

most and

where

in a reasonable

when

second

In an empirical

the

10 5

of n x 100 queen

of times

a basic

The

several

10 to one

n x 100 queens,

cases

"most-constrained

selects

moved,

be conducted moving

For the

results

n =

on a SPARCstationl)

and

from

A bound

the percentage

swamped.

10 4

Algorithms

program

of queens

after

> 12 hours

for n increasing

could

a solution

n =

for N-Queens

program.

Stone [23] found that this was by far the out of several described earlier by Bitner variable

required

hill-climbing

100 times

mean

10 3

of repairs

(100 runs

of our

run

in parentheses

backtracking

on which

687 (96%)

efficiency

n x 100 queen

first

backtracking

17.4

depending

did not

indicates

program

88650 (13%)

n =

of Backtracks/Repairs

was

shows

so that

program

credited

the

program

or a repair,

was employed If the

4473 (70%)

resources

2: Number

2 compares

backtrack

53.8

1: = number

computational

Table Table

n = 102

of backtracks,

• = exceeded

programs.

n = 101

n >

1000,

results

for

is a basic the

next

Stone

and

n-queens problem exhibited highly

815{ of the

100 backtracks.

success-

runs,

Unfortunately,

but

for

n > 1000,

one hundred

a SPARCstationl, the

both

Thus

in the

table

above,

this

of problem that used this

first"

we were shows

violations

this

never

results

are better

improvement) again

and

generating

time

a path

those

would

began

(i.e.,

had

for the hill-climbing

because

the

initiate

to increase

no queen

only

backtracking

significantly.

back-

next

row

As discussed

more

(although

program

when

the

However,

to be repaired

program

hill-climbing

about

because

each

The

heuristic.

on

program augmented

that

requiring

and

row after

for n > 1000.

the rain-conflicts

12 hours

rapidly

the next

data

than

regardless

heuristic

well,

grows

to select

sufficient using

more

50 repairs

along

primarily

of backtracks

O(n)

extremely

backtracked than

number

takes

considerably

row shows the results for an informed backtracking heuristic as described in the previous section. We

a pruning

of constraint

required

for hill-climbing

performed

with

mean

from

results

size. The final the min-conflicts

program

the

program

heuristic

prevented

the

algorithm

program

of the

because

"most-constrained

track.

runs

tends

for n >__100,

than

there

once).

is little

to repair

number

the

The

room

same

for

queen

again.

i°"

100

°p# °,

i

e_

i

o," o.°°°

._10 etp _

0.1

"

_..°ooO°°°_1

0.01

1

2

10

Figure We note

2: Mean that

for the

O(n) time in the number of repairs program runtime the

million

is less

than

any positions (note

queens

that

algorithm).

the

position

with

in less than

and

a half. with

the

zero:conflicts.

we already With

using

also be optimized

a minute

5

6

10

10

Program

on N-Queens

the min-conflicts

heuristic,

Problem

each

repair

requires

this is a relatively minor price to pay. Since the constant as n grows, the average runtime of the

linear. This is illustrated by figure 2, which shows the average program. In terms of realtime performance, this program solves

problem can

4

for Hill-Climbing

worst case. However, remains approximately

program

to find

Time

two programs

is approximately for the hill-climbing

This used

Solution

3

10 10 10 Problem Size

have

this

four

minutes

for large

Specifically, fewest

in the

conflicts.

To accomplish set

columns

necessary 8

in which

repair

The this,

of empty

this set, it is no longer

on a SPARCstationl.

problems, first

phase step

from

the the

the

a two

checks

a set of empty

to search

case

step

time

process

to see if ther_

columns

initialization entire

solution

is are

is maintained phase

of the

row for a zero-conflict

position. Only positions that are in an empty column could possibly empty columns is very small compared to the total number of columns for a suitable candidate in much less time. Second, that

if there

are

no positions

has only one conflict.

of a column diagonals

being are

columns

which

(with

replacement) it is very

compared

to the

choosing

one from

just

a few tries. we could

search

row

the to the

4.2

Since,

problem

involves constraints,

earlier,

is a complex

techniques

are

a constraint of the

The uled

approved number

by the

There may aration

is small

then

randomly position

that

the

program

could

of steps

and

number We never position

on a time

etc.

space

The

bothered

is quickly

line,

not then

to add

found.

subject

telescope

traditional

The

maximize

As mentioned

system,

the

system

has

is a set of detailed These

a peer

exposures

review

performed

process,

quite

specifications An

was

exposure

well using

data

be considand

that

be specified may

by the

proposer

be important. for

in the

of properties

others.

case

of periodic

light. Exposure Some exposures

relationships

[astronomer].

Some Some

and

exposures

must

Their

relative

are designed

be executed

phenomena.

among

scheduling

to augment simple

are to be schedproposals

includes

Some

at are

order

Johnston

exposures and

as calibrations

specific especially

times,

have

a potentially

is to be taken.

these

the

of the problem; to schedule one

a relatively

whose

specification

how the

can number

developed

by astronomers

describing

operations

an initial

for exposures

are submitted

parameters

that

problem,

and

problem both

earlier,

SPIKE,

to temporal

scheduling

backtracking

poorly.

we must

are satisfied.

initial

are a variety

or scattered intensity.

violation. here

problem:

acquisitions phases

that

of configuration

[13] outlines

row

a one-conflict

of tasks

or perform where

constraint-based

the

to SPIKE

large

position.

a one-conflict

of the

finds

exists

conflicting

1/4

constraint and

both

in the

of work

positions

generally

that

positions

the use of AI techniques highlighted the difficulty system, called SPSS, would take over three weeks

The

telescope.

been

test

amount

all such

on which

inapplicable problem

constraints

replace)

input

on the

the

a constant

preferences,

either

without that the

of observations.

(and partially approach.

probability

has only a single

after

a set

problem

optimization

system developed it was estimated week

that

for a position the probability

approximately

randomly

possibility

in practice

as the

we expect

can

program

the

placing

resource

importance

Since

searches of queens,

same

a position,

search

program

Applications

as discussed ered

the

for a minimum since

the

in identifying

random

constraints, research

such

theoretically,

this

however,

Scheduling

A scheduling

involved In practice,

completely

program,

find

set.

stop

the

distribution

program

is encountered

work

that

or 1/4.

the

we will quickly

O(n)

terminate, this

a position

conflicts,

is roughly

x 1/2

positions,

until likely

zero

a homogeneous

position

is 1/2

to be one-conflict

Since

after

a one-conflict

open,

with

If we assume

qualify. This set of and can be scanned

that

time

sep-

or target

or at specific

sensitive

durations may vary depending on background must be executed without interruption while

to stray light others

can be broken up as needed. In somecasesa specificorientation of an instrument aperture is required. Someexposuresare conditional on the results of other exposures. In addition to proposer-specifiedconstraints,there are a large number of other constraints from

that

"strict"

must

be considered

constraints

that

when

cannot

scheduling

HST

be violated

under

"good operating practices" that represent scheduling point closer than 50 ° to the sun and 15 ° to the bright is relatively spent

slow

(90 ° in -,_ 15 minutes)

in maneuvers.

altitude

(500

occulted

by

Many

kin) the

and

earth

collected.

a high

Scattered

particle

earthlight

of HST's

result

orbital

of each

period.

orbit.

Up

by HST's

density

changes

circumstances,

are

a direct

during

dramatically

over

to

the low

A typical to half

passage

region

range

HST is not allowed to Slewing the telescope to minimize

for up to ,v 20 minutes

Anomaly,

any

goals. moon.

They

so it is important

95 minute

for --, 40 minutes

day are contaminated Atlantic

constraints

consequent

operations.

the

the

orbits the

data

course

orbital

target

through

which

time is in a South

cannot

be

of an orbit...

The scheduling team at the Space Telescope Science Institute made the problem considerably more tractable by breaking it into two parts: the long-term scheduling problem and the short-term one year's days

scheduling

worth

length.

problem.

of exposures,

The

The

and

short:term

long-term

dividing

problem

problem

them

consists

up into

of coming

consists "bins"

of taking or time

up with

a very

approximately

segments

detailed

of a few

schedule

for

a time segment, which can be translated into commands that the telescope can then directly execute. Currently SPIKE handles only the long-term problem. The short-term problem has a quite

different

planning

nature,

to refer

goals, and the The short-term

because

to the

it involves

generation

system,

however,

will hopefully it can

here

may

contribute

SPIKE

operates

relative

generate

constraint

straint

that

functions

job.

a Schedule to this

scheduling.

(We use the to achieve

term

a set of

relations

et al.

[21] are

possibility

for significantly the exposure

developing is the

smaller

by improving

in question. the

are represented

the

of time between the desirability

the telescope when

specify

should target

the

AI planning

extension

relative

For example, not point

internally

before/after

one near

to the

of the

suitability the moon.

as piecewise 10

that

system

research

reported and

of tasks,

and

the

function is a function at a specified time,

function

may

represent

the suitability

functions,

so

system.

by astronomers

will be low (perhaps

constant

techniques SPIKE

the specifications into "suitability functions".

ordering

Thus,

The

SPIKE

prepared

suitability an activity

moon

of the

time buckets.

speed

specifications

tasks. Each of starting

is close

line.) such

instruments on the telescope, pointing maneuvers, the short-term problem is handled by the original

Another

goal,

by taking

amount represents

by the

an exposure

and

set of activities

they are internally consistent. It then compiles represented as relative temporal relations and

temporal

mal/minimal whose value

Muscettola

do a better

that

idating that of constraints,

planning

term scheduling to refer to the process of placing activities on a time problem requires planning because an exposure may require activities

as warming up or cooling down different communication of data, etc. Currently, SPSS

both

of a partially-ordered

zero).

enabling

vala set The

maxi-

of time as given the

con-

of scheduling Suitability combinations

of multiple suitabilities to be calculatedefficiently. Becauseof the uncertainty inherent in some constraints, and also becausethe grainsize of the time segmentsmay be relatively large, suitability functions are often used to represent

the

statistical

time

segment.

must

be taken

the earth's the

For

many

Once the

has

In other

compiled

Discrete

The

the

asymmetrically to an auxiliary that the network can assume. for many

disadvantage case

the

which

network

it will

number

problems,

is that

can fall into

oscillate.

of neuron

To

illustrate

state

is used

or block

of exposures)

segment

means

the

tasks,

connections

where each

the

task

of neurons weights

between

suitability

representing input

a task,

is updated

and

The

flipping

network

in conflict

then

network

the

the

network

Soft

constraints

state

the "least

by randomly neuron

hard

suitability conflicted"

set

neuron in that

set

accomplishes

schedule,

and

segment

that

constraints

11

if the has

zero

to move

neuron

will provide

the

than

task

with

one

neuron.) represents

inconsistent their

If the task

is currently

set

connection

that

is most

consistent

following:

between To insure

for each

way

input

it

negatively

neuron

on more

input,

is currently

unscheduled

then

constraint

violations.

Note

it treats

suitabilities

as zero

and

are

the fewest

(i.e.

is between places

the

is "on"

(i.e.,

a set of neurons are

the

(an exposure

is zero.

guard

whose

states

some

how

constraints

to the

will turn

among

after

if a neuron

hard

(Due

selecting

all neurons'

states

for each possible

Inhibitory

is on, the

The

in which

us consider

is a guard

on.

a

over. let

configuration

there

one

the guard

the

to indicate

in the

machine.

guaranteed,

or "off";

used

to turn

cycle

time

"on"

segment

is coupled

the configurations to rapidly find

one neuron

segment.

search,

is a modified

network

task to be scheduled

time

in

it must

this

to converge

scheme,

in a certain

that

roughly

from

it for the

only represents (where

a time

When

scheme

it is removed

are

two tasks

of the

(if any).

started

is either

neurons

enough

and

Each

out

of unstable

fails

could

occur.

on a serial

a group

is in

the exposure

network main

restricts network

set of neurons,

that

if no neuron

is large

updating

are two or more

for

assigned

on each

schedules

that there

the

are set up, it is unlikely

network

neuron

of placing a task;

that

with its current output solution is achieved.

the

Each scheduled

is eventually

an excitatory The

schedule.

is currently

problem.

(which

of constraints,

GDS the

network

updating

by a separate

segment,

is no longer

stopped

and

the HST scheduling

The

involving

it is simply

time

to carry

is simulated

if the

for each segment

a set

is that

configuration

and

opportunities

neurons which enables the

minimum

architecture

is represented

in the task

weighted) that

to solve

14].

network

however,

transitions,

network

the

to a stable a local

into

a certain

an exposure limb

to schedule

network

modification

that

earth's

that

viewing

a neural

network[2,

when

indicate, over

proposals

employs

In practice,

the

network time

even

might

during

state the

be preferable

network of guard This modification

convergence

5 ° from

of such

astronomers'

significant

might

satisfied

it would

(GDS)

most

are

high number

SPIKE

Stochastic

network[10].

solution

words,

a relatively

than

function

conditions

an exposure

constraint

more

suitability

these

schedule.

of scheduling

orbital

is pointing

resulting

of time

a good

Guarded

Hopfield

The

in which

SPIKE for

desirability

a particular

telescope

orbits).

segment

search

the

amount

encompass a time

when

shadow.

average

or aggregate

example,

a task.

one)

only

consulted

or one). when

a

The

rain-conflicts

algorithm

on representative

data

the min-conflicts was

developed

from

application,

the

an

the

analysis

(or the

process

time,

is that

advantage

performance.

much

of the

overhead). could

Moreover,

be quickly

an order

such

difficult.)

and

this

currently

experimenting

with

min-conflicts

the

the improvements since

the

Several

minor

non-binary

constraints,

constraint

assignment HST

with

application

a variety

an implementation was

of different

Although

this

search

observed

will eventually

a larger

number

i.e.,

when with

constraints

the

number

CSPs,

the

depend

on the

it sufficed

implementing binary

that of tasks

exact each

may

involve may

violated

and

then

contrast, them perhaps

and

As described reschedule

earlier,

the

task

the min-conflicts reinserting when

the

them schedule

the network in two separate

algorithm later.

the

steps,

rearranges

tasks

It appears

that

is over-constrained

on the

8We discovered the importance of a good initial assignment it has also been shown to hold for the network as well.

12

not

algorithm, includes

For

example,

a given

of conflicts even

and

the

task

from

an

for the though

min-confllcts the

schedule

consecutively.

rather

is not

time for

out,

conflict,

occur

schedule,

difference

(as discussed

schedules,

the

As it turned

network may

that

problem

during

number

a conflicted

which this

First,

as a single

GDS

will remove

be combined

better

variables.

in question.

constraint

that

We are

we expect

into

scheduled the

is due

comparison

can

scheduling

several

be

of counting

constraint

multiple tasks might be involved in the violation. A second issue concerns a difference between algorithm.

HST

at least

schedules.

application.

The

that

method

particular

to count

the HST

program

runs

factor.

that

translate

space

we believe

complete,

of acceptable

constraints.

network,

improvement

enabling

yet

trying

the

a precise

strategies

is not

of the

randomly

GDS

program

issue,

a significant

study

than

the scheduling

of the

as just

analysis

(particularly

makes

method

we have

arose

For general may

of the

some which

is

at each point

scheduling

although

differences,

in

schedule improve

to the

is so simple, (The

can explore

3, deals

bounds

network,

be regarded

heuristic.

issues

in section

segment.

may

process

as specified one

the

rather

schedule,

can be eliminated

can In the

schedule

our

greatly

as compared

heuristic

approaches

Because could

schedule,

on the

efficient.

language

formulation

in speed

search

network

is extremely

than

task

algorithm

machines.)

an initial

bound).

network In effect,

the

a conflict-free

assignment

algorithm,

the min-conflicts

in C and faster

simple

the

until

an initial

each

min-conflicts

as programming

While

clear

initial

to create

two

on serial

by constructing

schedule iteration

a good places

of using

because

coded

of magnitude

to factors the

overhead

the

by a preset that

algorithm

the

operates

repairs

In fact,

(The

are run

GDS

Institute.

network.

both

algorithm

as the

Sciences

GDS

currently

showed

in using

Telescope

of the

network's

assigning tasks. 5 The greedy algorithm to minimize the number of conflicts. One

as effective

but

is terminated

we use a greedy

Space

behavior

iteratively

algorithm

to be at least

of the

manner,

and

shown

by the the

min-conflicts

phase,

min-conflicts

solution

mimics

in a similar

a preprocessing found

provided

algorithm

be parallelized HST

sets

has been

than

significant,

In

removing except

below).

by analyzing

the min-conflicts

algorithm,

but

4.2.1

The

The

HST

Over-Subscription

scheduling

we must

problem

maximize

fied

[8, 20].

that

many

We note

Unfortunately, objective

both

more

can

the

the

both

the overall

be accommodated

has

yet

been

established

by the

the

problem

to its users.

There

in conflict,

and

network

tasks.

SPIKE's

(Unscheduled

will be moved

off the

approximately

(assuming

can be removed. to try

One

a variety

experimenting network and

to the

(where

use

a more

schedule,

place

Other

The

min-conflicts

with

good

are removed

currently tic

and/or

re-inserted), alter

found)

that

tasks

are

any

tasks

but vary

the probability

likely

to remain

is eventually

that

are

conflicting

number

tasks

of conflicting

taken

the procedure

after

tasks

coming

by the

for removing an unscheduled tasks. (If we but simply be

Another

need

in conflict

it is relatively

We are currently

of choosing

case.)

of these

in conflict

is that

problems.

or

are

algorithm

algorithm

as in the normal

for removing

GDS

and

Pearl

3-colorability,

an undirected to the found graph.

graph

constraint that

poorly,

graphs

the

the

becoming

network

types

have

with

conjunctive

n vertices.

connected

approach

up with

is to

an initial

to be removed.

of the GDS vertex

vertices network (with

in local minima rapidly

generated

our method network

NP-complete Each

and

problem. must depended vertex

problems described

problems[19].

We are

well.

the min-conflicts

heuris-

In this

we are

be assigned

be assigned

average

of other problems

works

greatly degree

problem, one of three

the same

with high probability.

to a solution.

13

on a variety matching

for which

of the graphs

tried

randomly

precondition

no neighboring

performance

converged

the

of applications

a well-studied

caught

also been

including

the performance

that

On sparsely

formed

network

results,

[6, 2] and the

also compared

of the

later

we can

the minimum

preliminary)

cataloging

Johnston

and

on the schedule

method

only

on graph

subject

been

to select

tasks

the

quite

unscheduled

is to follow the approach

first

that

two goals

apparently

scheduled task, or bound the number of unscheduled then tasks will never be removed from the schedule,

We have given

When

with overconstrained The

like

Applications

(but

by Dechter

for dealing

these

but

likely and

of the min-conflicts

approaches.

of proposals

are either

of unscheduled

has not

no clear

we would

between

ad-hoc,

schedule,

schedule

advantages

For example,

principled

4.3

the number

in

schedule.

is that

number

that

is equally

the

a conflict-free

to place

so that

Thus,

two basic

tasks.

from

scheme onto

satis-

Institute.

is a bit

in conflict.

task versus an already set the bound to zero, moved

updating

by any

scheduler

the

where

are

over-subscribed,

In particular,

tradeoff

of tasks

of tasks

of the

tasks

inserting

a pool

number

of schemes

with

that

is, in effect,

will be moved

schedule.)

equal

interrupted easy

tasks

and the

Science

in a manner

satisfactory

of the cases.

schedule

for evaluating Telescope

highly

that

be accommodated

in such

of the

policy Space

can

problem

constraints

to remain

than

schedule

optimization

of the

the performance

best

suitability

- no clear

handles

is expected

in analyzing the

a constraint

the importance

telescope

for determining

can

and

will be submitted

one difficulty

to maximize

be considered

number

that

proposals

exists

SPIKE

Problem

color.

Adorf

colors and

on the

connectivity

4) the

network

On densely

per-

connected

We have

repeated

found

similar

using

the

Adorf

results.

that

the

exceeds

experiments

also experimented

heuristic.

records

of backtracks

Johnston's

We have

min-conflicts

program

and

Our

assignment

most

with

a (dynamically

with

with

effective

the least

adjusted)

our hill-climbing

variations program

conflicts

and

of informed

backtracking

is an informed

backtracking

found

threshold,

program,

so far.

the

When

search

the

process

number

is restarted

using this best assignment. We have found that performance is further improved by adding heuristics for selecting which vertex to repair, and that, as in the n-queens problem, it helps to have This

a good

initial

illustrates

performance

domain,

certain

Brelaz's

constrained has

shownthat

and

this

can also that

be produced

combining

all of the

the

GDS

are

using

multiple

additional

heuristics.

heuristics

can

improve

color

in the

to produceexcellentresults.

two strong

informed

will optimally

discussed

known

[5] employs our

heuristics

backtracking "almost

For

(forms

algorithm.

all"

random

of "mostTurner

k-coIorable

[25]

graphs

is not surprising.

of Experimental

tasks

network

algorithm

it outperforms so its dominance

Summary

For

heuristicmethods

algorithm

backtracking,

4.4

which principle

k-colorability

first")

without

assignment, well-known

significantly.

In this instance,

the

Results previous

can be aiSproximated

section,

we have found

by hill-climbing

wiih

the

that

the

min-conflicts

behavior

of

heurlstic.

To

this extent, we have a theory that explains the network's behavior. Obviously, there are certain practical advantages to having "extracted" the heuristic from the network. First, the

heuristic

is very

simple,

and

done in a task-specific manner. differen(search strategies _ and practical

applicati6ixs.

have

previously

......

successfully

been

:::-

also

scheduling

considered

extremely

especially

if

: = _:-

Furthermore,

experimental results cl-eariyoutperirorms

the heuristic

problems.

variations

efficiently,

can then be used in combination with Thls is a s_gnl_cazit factor for most

heuristic is conc6_erned,our the m_ic-/s_euHsti-c

investigated.

to real-world

We have

be programmed

Second, the heuristic task-spec-i_c-_e=ur_stlcs.-

Insofar as the power of the aging. On t_e n-queens-pr0biem that

so can

has

-=_-

of the

min-conflicts

already

are encourheuristics been

_

heuristic,

applied -

such

-

as repairing

the

.......................

variable prove

that the

changed_ sistent,

5 The

participates

in the

performanace

most

of the

algorithm,

As long as-the_euristic

tends

it appears

that

our

basic

results

conflicts and

first.

to decrease tend

In some

in other

cases

t_e

cases,

performance

number

such

variations

is not

of variables

im-

significantly

that

are incon-

to hold.

Analysis p_rev!pus

section

such

as placing

tasks, coloring influence

showed queens

sparsely connecte the effectiveness

that on

the

min-conflicts

a chessbo_d,

heuristic and

less

is extremeI effectiveon

d graphs. In this section, we analyze of the heuristic. Consider a CSP

14

X effective:on other

tasks,

some such

as"

how the parameters of a task with n variables, where each

:

variable

has

every

variable

a single the

k possible is subject

solution

following

a mistake

to the

that

our informed

backtracking

an exponential

amount

For

any

values

a value

is, exactly

that

to a variable a mistake

of values

to the

the

model

where

there

is only

assignment.

solution

of this sort may from

We address

heuristic

will make

We define is found.

prove

a mistake

We note

fatal,

as

that

as it may

for

require

a mistake.

variables,

to convert

the

that

min-conflicts

is in conflict?

before

to recover

we assume

satisfying the

that

to a simplified

and

one

probability

program,

to be changed

consideration

constraints,

to be changed

of search

assignment

will have

that

is the

will have

our

c binary

problem, What

it assigns

a value

We restrict

to exactly

question:

when

choosing

values.

there

will be a set

assignment

into the

of d variables

solution.

whose

We can regard

d

as a measure of distance to the solution. The key to our analysis is the following observation. Given a variable V to be repaired, only one of its k possible values will be good s and the other

k -

at most other

1 values d other

variables.

a mistake then

Thus,

when

the

will be bad variables

this

We can

V. since

idea

variables above,

above that

the

value

share

is greater

bounding

the

the

a constraint

than

We can,

probability

that

each

of which

of Nb is pc.

probability is either

suppose

be less likely

has more the than

than

therefore,

good

probability if the the

or equal

to make

value

heuristic

there

of a conflict

number

as c

when

value. will make

to V by a constraint. for V' with probability

of Nb is pc, because

bound

Nb is less than

at most

the

a mistake

with

d conflicts,

any bad

min-conflicts related value

conflict as many

Nb be the variables. exactly

c

is p. As mentioned

of conflicts

probability

are

We p,

Nb for each

of making

a mistake

bad by

to d.

inequality, which states that the sum N of n indepenvariables is less than the expected value/_" by more than

e -2_2".

In our model,

1 or 0, depending

on whether

Nb is the sum of c potential there

is a conflict.

The

conflicts,

expected

value

Thus:

we are that

should

conflicts

the

may with

- it will select fewer

that

value

V, and

Pr(Nb Since

probability

make

To bound Nb, we use Hoeffding's dent, identically distributed random sn only with

a mistake will have

expected

with

will not d.

heuristic

value

conflict

of the k - 1 bad values

value

the

good

may

V and V'. Consider an arbitrary bad value for V. Let between this bad value and the values for the other

assumptions,

min-conflicts

min-conflicts make

good

the

value

variable V. Let V r be a variable for V conflicts with an arbitrary

independent of the variables total the number of conflicts the

Whereas a bad

if each

cannot the

to bound

a mistake when repairing assume that a bad value

Given

the

In fact,

heuristic

variable,

use this

mistakes).

assignment,

as d shrinks,

it repairs

min-conflicts

it repairs

(i.e.,

in the

interested

d is less than

in the

_< pc -- sc)