Ames. Research. Center. Artificial Intelligence Research Branch. Technical ... B. Philips. _. Mark. D. Johnston. 2. Philip. Laird 3. 1Sterling Federal Systems ...... (1+ (eli. *numcolelts* col). )) (setf. (elt. _numdndiagelts*. (+ limit. ( row col))). (1+.
?',j f_
The
MinConflicts
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
24417
CENTEI_ 94035
N_225445
MINCONFLICTS AND THEORETICAL
p Unclas O091blO
G]/6]

_A._A
Ames Artificial
Research Intelligence Technical
Center Research Branch Report
FIA9125
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
hillclimbing
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
minconflicts
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 NQueens
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 MinConflictsHillClimbing$ 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.
threequarters
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
nqueens
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
"mostconstrained
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 NQueens
program.
Stone [23] found that this was by far the out of several described earlier by Bitner variable
required
hillclimbing
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
nqueens 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 hillclimbing
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
hillclimbing
about
because
each
The
heuristic.
on
program augmented
that
requiring
and
row after
for n > 1000.
the rainconflicts
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 hillclimbing
performed
with
mean
from
results
size. The final the minconflicts
program
the
program
heuristic
prevented
the
algorithm
program
of the
because
"mostconstrained
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 NQueens
the minconflicts
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 HillClimbing
worst case. However, remains approximately
program
to find
Time
two programs
is approximately for the hillclimbing
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 zeroconflict
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 oneconflict
of tasks
or perform where
constraintbased
the
to SPIKE
large
position.
a oneconflict
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 oneconflict
Since
after
a oneconflict
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 proposerspecifiedconstraints,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 longterm scheduling problem and the shortterm one year's days
scheduling
worth
length.
problem.
of exposures,
The
The
and
short:term
longterm
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 longterm problem. The shortterm problem has a quite
different
planning
nature,
to refer
goals, and the The shortterm
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 shortterm 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 partiallyordered
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
rainconflicts
algorithm
on representative
data
the minconflicts 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
minconflicts
the
the improvements since
the
Several
minor
nonbinary
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 minconflicts reinserting when
the
them schedule
the network in two separate
algorithm later.
the
steps,
rearranges
tasks
It appears
that
is overconstrained
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
minconfllcts 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 conflictfree
assignment
algorithm,
the minconflicts
in C and faster
simple
the
until
an initial
each
minconflicts
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
minconflicts
phase,
minconflicts
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 minconflicts
algorithm,
but
4.2.1
The
The
HST
OverSubscription
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
minconflicts
with
good
are removed
currently tic
and/or
reinserted), 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
3colorability,
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
NPcomplete Each
and
problem. must depended vertex
problems described
problems[19].
We are
well.
the minconflicts
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 wellstudied
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 minconflicts
approaches.
of proposals
are either
of unscheduled
has not
no clear
we would
between
adhoc,
schedule,
schedule
advantages
For example,
principled
4.3
the number
in
schedule.
is that
number
that
is equally
the
a conflictfree
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
oversubscribed,
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
minconflicts
program
and
Our
assignment
most
with
a (dynamically
with
with
effective
the least
adjusted)
our hillclimbing
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 nqueens 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
kcoIorable
[25]
graphs
is not surprising.
of Experimental
tasks
network
algorithm
it outperforms so its dominance
Summary
For
heuristicmethods
algorithm
backtracking,
4.4
which principle
kcolorability
first")
without
assignment, wellknown
significantly.
In this instance,
the
Results previous
can be aiSproximated
section,
we have found
by hillclimbing
wiih
the
that
the
minconflicts
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 taskspecific manner. differen(search strategies _ and practical
applicati6ixs.
have
previously
......
successfully
been
:::
also
scheduling
considered
extremely
especially
if
: = _:
Furthermore,
experimental results cleariyoutperirorms
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_euHstic
investigated.
to realworld
We have
be programmed
Second, the heuristic taskspeci_c_e=ur_stlcs.
Insofar as the power of the aging. On t_e nqueenspr0biem that
so can
has
=_
of the
minconflicts
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 asthe_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
minconflicts
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
minconflicts
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
minconflicts 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,
minconflicts
minconflicts 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
minconflicts
it repairs
(i.e.,
in the
interested
d is less than
in the
_< pc  sc)