Model Presolve, Warmstart and Conflict Refining in CP Optimizer

© 2013 IBM Corporation
Sept. 16, 2013
CP-2013 Workshop on "CP Solvers: Modeling, Applications, Integration, and Standardization"
Model Presolve, Warmstart and
Conflict Refining in CP Optimizer
P. Laborie, P. Refalo, P. Shaw
IBM Software Group

Overview
 A “model-and-run” optimizer must provide:
– Efficient & robust automatic search
– Easy-to-use functionalities to control the search
– Productivity tools for model development

Overview
CP Optimizer is a “model-and-run” optimizer providing:
Restart, Learning, Portfolios, Relaxations, LNS, Presolve, …
Parameters, Search phases, Warmstart, …
Development Studio, Search log, Conflict Refiner, ...

Model Presolve
 Depending on the input data, models can have redundancies and
can contain poorly formulated constraints
 Objective: Automatically improve the model in order to make
stronger inferences faster
 Simplifications
–Constraint compaction
–Redundancy elimination
–Constant propagation
–Common sub-expression factorization
 Aggregations
–Count expressions & difference constraints
–Linear constraints over binary {0,1} variables
–Alternative constraint combined with arithmetic expressions

Examples of Model Presolve
 Elimination of redundant constraints
– Bound of variables and expressions computed by constraint
propagation are used to eliminate redundant constraints before
search
 Propagation of constant variables and expressions
1 + 2x + 3y + xy – 3x + 7y ≤ 8x – 7y + 10
x == 4 → 21y ≤ 45
 Variable merge
x==y, y==z, z==t → merge the domains of x, y, z and t
and replace y, z, and t by x
everywhere

 Aggregation of difference constraints
– Binary ≠ constraints can be lifted to alldiff constraints
A problem with ≠
x ≠ y
x ≠ z
x ≠ t
y ≠ z
y ≠ t
z ≠ v
t ≠ z
t ≠ v
u ≠ v
alldiff(x, y, z, t)
alldiff(z, t, v)
u ≠ v
The associated graph structure
The aggregated model
y t
z u
v
x
≠
≠
≠
≠
≠
≠
≠
≠
≠

 Lifting difference constraints
1) Pick up a binary constraint x ≠ y not already in a lifted alldiff
2) Find the largest clique containing x ≠ y (in practice we use a
greedy algorithm – add vertex one by one to the set starting
with vertices having the maximum degree)
3) Add a new lifted alldiff
 Experimentation on graph coloring
– Color the vertices of a graph such that two adjacent vertices
have a different color and the number of colors is minimized
– The model is stated with a set of binary ≠ constraints
– Problems are from the DIMACS challenge (COLOR02 set)
• 9 problems over 60 are solved to optimality without presolve
• 39 problems over 60 are solved to optimality with presolve

Results on graph coloring problems
Problem Time Ch. Pts Time Ch. Pts max
anna 1000.01 19308059 0.08 138 11
david 141.49 3628877 0.06 87 11
fpsol2.i.1 1000.01 6353782 4.11 496 65
fpsol2.i.2 1000.01 8731886 2.03 451 30
fpsol2.i.3 1000.01 8980436 1.90 425 30
games120 1000.01 20346758 0.08 120 9
homer 1000.02 10225244 0.05 0 13
huck 1000.01 28282739 0.05 74 11
inithx.i.1 1000.02 4744285 14.18 864 54
inithx.i.2 1000.01 5741171 10.06 645 31
inithx.i.3 1000.01 5671424 10.07 621 31
jean 1000.01 26779859 0.05 80 10
le450_15a 1000.02 3623748 8.41 1062 15
le450_15b 1000.02 6650123 4.70 847 15
le450_15c 1000.01 5723362 1000.82 630981 15
le450_15d 1000.02 1457738 1000.79 49765 15
le450_25a 1000.01 7289172 3.52 450 25
le450_25b 1000.01 7355812 3.25 450 25
le450_25c 1000.02 1969772 1000.58 45443 25
le450_25d 1000.01 2313797 1000.67 52443 25
le450_5a 1000.02 3009929 1000.37 114323 5
le450_5b 1000.01 2379044 1000.37 107031 5
le450_5c 1.48 522 10.17 450 5
le450_5d 1000.02 1635272 1000.75 69870 5
miles1000 1000.01 11725830 0.47 128 42
miles1500 1000.01 9798720 1000.02 136613 71
miles250 0.42 6961 0.05 128 8
miles500 1000.01 15628706 0.13 128 20
miles750 1000.01 13420578 0.27 128 31
mug100_1 1000.01 20055287 5.95 91443 3
mug88_1 1000.01 22220677 5.44 92818 3
mug88_25 1000.01 22069362 1.19 20104 3
Problem Time Ch. Pts Time Ch. Pts max
mulsol.i.1 1000.01 13151679 0.66 197 49
mulsol.i.2 1000.01 16062965 0.45 188 31
mulsol.i.3 1000.01 16000296 0.45 184 31
mulsol.i.4 1000.01 16442846 0.44 185 31
mulsol.i.5 1000.01 11138968 0.59 186 31
myciel3 0.02 32 0.02 32 0
myciel4 0.09 2791 0.09 2791 0
myciel5 234.11 5734555 239.05 5734555 0
myciel6 1000.01 15070869 1000.00 14612561 0
myciel7 1000.01 10789203 1000.00 10699137 0
queen10_10 1000.02 7167824 1000.04 650963 10
queen11_11 1000.01 5350496 1000.04 564951 11
queen12_12 1000.01 4302509 1000.04 357267 12
queen13_13 1000.01 3613703 1000.08 259550 13
queen14_14 1000.01 3023318 1000.10 276221 14
queen15_15 1000.02 2544350 1000.15 180290 15
queen16_16 1000.02 2114831 1000.15 166693 16
queen5_5 0.02 45 0.02 25 5
queen6_6 1.44 24392 1.45 7726 6
queen7_7 1.28 19528 0.09 111 7
queen8_12 1000.01 8415680 1000.04 890550 12
queen8_8 1000.01 11742448 1000.03 1826017 8
queen9_9 1000.01 9198487 1000.03 724638 9
school1 1000.02 1938387 28.52 543 14
school1_nsh 1000.02 4461519 1000.87 251760 14
zeroin.i.1 1000.01 11703893 0.55 211 49
zeroin.i.2 1000.01 11807063 0.47 211 30
zeroin.i.3 1000.01 12225170 0.50 206 30

 Common sub-expression factorization
–Eliminate multiple occurrences of the same expression and
replace them by a new variable
example: xy ≠ z + t
z + xy == a + b
100 ≤ z + xy
–The expressions xy and z + xy appear several times. We
introduce two new variables u and v to replace these
expressions and add the constraints: u == xy, v == z + u
–The model becomes u ≠ z + t
v == a + b
100 ≤ v
–Communication of bound reduction on newly introduced
variables achieves more domain reduction
–It also reduces the number of expressions and thus involves less
computations

Examples of Model Presolve: Golomb Ruler Example
using CP;
int N = 10;
int L = ftoi(pow(2, N-1) - 1);
range R = 1..N;
dvar int x[R] in 0..L;
minimize x[N];
subject to {
forall(i in 1..N-1) x[i] < x[i+1];
forall(i in 1..N-1,
j in i+1..N,
k in i..N-1,
l in k+1..N : l != j){
x[j] - x[i] != x[l] - x[k];
}
}
let x0 = intVar(0..26);
let d1 = intVar(-25..27);
let d10 = intVar(-25..27);
let d11 = intVar(-24..28);
let d12 = intVar(-23..29);
let d13 = intVar(-25..27);
let d14 = intVar(-24..28);
let d15 = intVar(-25..27);
minimize(x5);
alldiff([d1, d2, d3, d4, d5,
d6, d7, d8, d9, d10,
d11, d12, d13, d14, d15]);
x0 < x1;
x1 < x2;
x2 < x3;
x3 < x4;
x4 < x5;
d1 = x1 - x0;
d2 = x2 - x0;
d3 = x3 - x0;
d4 = x4 - x0;
d5 = x5 - x0;
d6 = x2 - x1;
d7 = x3 - x1;
d8 = x4 - x1;
d9 = x5 - x1;
d10 = x3 - x2;
d11 = x4 - x2;
d12 = x5 - x2;
d13 = x4 - x3;
d14 = x5 - x3;
d15 = x5 - x4;
Golomb ruler
naive formulation
(in OPL)
Common sub-expression elimination
regroups repeated differences
x[i] != x[j] and introduces a new
variable for each
This allow aggregation of binary
difference constraints on these
variables to a single alldifferent
Presolve produces the well-known
tight model for this problem
Internal formulation
after presolve

Conflict Refiner
 Objective: Identify a reason for an inconsistency by providing a
minimal infeasible subset of constraints for an infeasible model
 Use cases:
1) Model debugging (errors in model)
2) Data debugging (inconsistent data)
3) The model and data are correct, but the associated data
represents a real-world conflict in the system being modeled
4) You create an infeasible model to test properties of (or extract
information about) a similar model

Conflict Refiner
 Objective: Identify a reason for an inconsistency by providing a
minimal infeasible subset of constraints for an infeasible model
 Use cases:
1) Model debugging (errors in model)
2) Data debugging (inconsistent data)
3) The model and data are correct, but the associated data
represents a real-world conflict in the system being modeled
4) You create an infeasible model to test properties of (or extract
information about) a similar mode

© 2013 IBM Corporation13
 USAF Satellite Control Network scheduling problem [1]
 n communication requests for Earth orbiting satellites must be
scheduled on a total of 32 antennas spread across 13 ground-
based tracking stations
 In the instances, n ranges from to 400 to 1300
[1] Kramer & al.: Understanding Performance Trade-offs in Algorithms for Solving Oversubscribed Scheduling.
Conflict Refiner example: satellite scheduling problem

Station1Station2Station3

Communication requestsTaski

Taski
Oppi,1
Oppi,2 Oppi,3
Alternative assignments
to stations × time windows
(opportunities)
Oppi,4
OR

Oppi,1
Oppi,2
Oppi,4
Oppi,3
Selected opportunity will use
1 antenna for communication
with the satellite
Taski

Conflict Refiner example: model 1

Conflict Refiner example: running model 1
! ----------------------------------------------------------------------------
! Satisfiability problem - 2,980 variables, 851 constraints
! Workers = 2
! TimeLimit = 30
! Problem found infeasible at the root node
! ----------------------------------------------------------------------------
...
! ----------------------------------------------------------------------------
! Conflict refining - 851 constraints
! ----------------------------------------------------------------------------
! Iteration Number of constraints
* 1 851
...
* 11 3
* 12 1
* 13 1
! Conflict refining terminated
! ----------------------------------------------------------------------------
! Conflict status : Terminated normally, conflict found
! Conflict size : 1 constraint
! Number of iterations : 13
! Total memory usage : 10.6 MB
! Conflict computation time : 0.04s
! ----------------------------------------------------------------------------

 Conflict:
 Opportunities for task “373A”:

! ----------------------------------------------------------------------------
! Satisfiability problem - 2,980 variables, 851 constraints
! Problem found infeasible at the root node
! ----------------------------------------------------------------------------
...
! ----------------------------------------------------------------------------
! Conflict refining - 851 constraints
! ----------------------------------------------------------------------------
! Iteration Number of constraints
* 1 851
* 2 426
...
* 58 5
* 59 5
! Conflict refining terminated
! ----------------------------------------------------------------------------
! Conflict status : Terminated normally, conflict found
! Conflict size : 5 constraints
! Number of iterations : 59
! Total memory usage : 13.3 MB
! Conflict computation time : 0.51s
! ----------------------------------------------------------------------------

 Conflict:
 There is not enough antennas to accommodate all 4 tasks on their
time-window on ground station “LION” (3 antennas):
<"134A",6,1232,19,1266>
<"144", 6,1238,31,1272>
<"146", 6,1228,22,1260>
<"146A",6,1230,22,1262> 1232 1266
134A
12721238
144
1228 1260
146
1230 1262
146A

 Solution with 825 tasks executed for a total of 838 candidates
(98.4%)

Warmstart
 Objective: Start search from a known (possibly incomplete) solution
given by the user in order to further improve it or to help to guide
the engine towards a first solution
 Use cases:
1) Restart an interrupted search with the current incumbent
2) Start from an initial solution found by an available heuristic
3) Goal programming for multi-objective problems
4) When finding an initial solution is hard, solve an initial problem
that maximizes constraint satisfaction and start from its solution
5) Successively solving similar problems (e.g. dynamic scheduling)
6) Hierarchical problem solving (e.g. planning scheduling)→

 USAF Satellite Control Network scheduling problem [1]
 n communication requests for Earth orbiting satellites must be
scheduled on a total of 32 antennas spread across 13 ground-
based tracking stations
 In the instances, n ranges from to 400 to 1300
 Tasks have priorities: first maximize the number of scheduled high
priority tasks, then the number of scheduled low priority tasks
Warmstart example: satellite scheduling problem
[1] Kramer & al.: Understanding Performance Trade-offs in Algorithms for Solving Oversubscribed Scheduling.

1 2Steps:

Overview
CP Optimizer is a “model-and-run” optimizer providing:
Restart, Learning, Portfolios, Relaxations, LNS, Presolve, …
Parameters, Search phases, Warmstart, …
Development Studio, Search log, Conflict Refiner, ...
Questions ?

Model Presolve, Warmstart and Conflict Refining in CP Optimizer

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