Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Study of Graph Coloring Problems using
SAT-based approach
.
Indira Bhusal Pandey
.
Graduate School of Intercultural Studies
Kobe University, Japan
CSPSAT meeting, March 13, 2009
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Outline of this talk
Overview
Applications of GCP/B-GCP
Definitions of GCP and B-GCP
COP representation of GCP/B-GCP
SAT problems
SAT solvers and SAT-based approach
SAT encoding methods
Order encoding of GCP/B-GCP
Developed GCP/B-GCP solver
Performance evaluation
Conclusion
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Overview
Objective
Study and implementation of a SAT-based GCP/B-GCP solver.
SAT-based approach is a general method to solve difficult
combinatorial problems by encoding them into SAT
(Satisfiability Testing) problems and solving by using an
efficient SAT solver.
A (pure) Graph Coloring Problem (GCP) is a problem to find
the minimum number of colors (chromatic number) such that
no two adjacent vertices have the same color.
A Bandwidth-GCP (B-GCP) is a generalization of GCP in
which a positive minimum distance is given as the weight for
each edge.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
Applications of GCP/B-GCP
GCP/B-GCP are useful in variety of applications. Some of them
are as follows:
1
Frequency Assignment Problems
Resolving the conflicts in radio frequency assignment.
2
Time Tabling Problems
Determining the minimum number of time slots needed
subject to their restriction.
3
Scheduling Problems
Resolving a number of pairwise restrictions on which jobs can
be done simultaneously.
4
Register Allocation Problems
Finding an efficient register usage in compiler systems.
.
.
.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
Graph Coloring Problems (GCP)
.
1
Given:
A graph G (V , E )
V : a set of vertices
E : a set of edges
2
Objective:
to find a coloring function c : V → {1, 2, 3, . . . , k} with the
minimum (chromatic number) k
such that colors of any two adjacent vertices must differ in at
least by one.
.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
Bandwidth Graph Coloring Problems (B-GCP)
.
1
Given:
A weighted graph G (V , E , w )
V : a set of vertices
E : a set of edges
w : a weight function E → N
2
.
Objective:
to find a coloring function c : V → {1, 2, 3, . . . , k} with the
minimum (chromatic number) k
such that colors of any two adjacent vertices must differ in at
least by the given weight.
Note: B-GCP is a generalization of GCP.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
.
Example of B-GCP
1
Given:
A weighted graph G (V , E , w )
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
.
Example of B-GCP
1
Given:
A weighted graph G (V , E , w )
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
.
Example of B-GCP
1
Given:
A weighted graph G (V , E , w )
2
Three is the minimum chromatic number. The following is a
coloring function with the minimum chromatic number.
.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Applications GCP
COP representation of GCP/B-GCP
B-GCP (and also GCP) can be formulated as a constraint
optimization problem (COP) on integers.
G (V , E , w )
Vertex vi
Edge {vi , vj }
with weight c = w ({vi , vj })
COP
Variable xi ∈ {1, 2, 3, . . . , k}
Constraint |xi − xj | ≥ c
Objective: minimize k
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
SAT Problems SAT solvers SAT encoding methods
SAT Problems
SAT (Satisfiability Testing) is a problem to decide whether
the given propositional formula is satisfiable or unsatisfiable.
The formula is given as a Conjunctive Normal Form (CNF)
A CNF formula is a conjunction (AND) of clauses.
A clause is a disjunction (OR) of literals.
A literal is either a Boolean variable or its negation.
A formula in CNF is satisfiable iff there exists an assignment
for each Boolean variable so that the whole formula becomes
true.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
SAT Problems SAT solvers SAT encoding methods
SAT Solvers and SAT-based approach
SAT solver is a program to find a solution of a SAT problem.
Recent advances of SAT solver technology are remarkable.
MiniSat solver (Eén and Sörensson 2003) can solve a problem
with 107 literals.
SAT solvers are used to solve hard problems by encoding them
to SAT problems (SAT-based approach), such as
Scheduling,
Planning, and
Software and hardware verification.
There are several SAT encoding methods.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
SAT Problems SAT solvers SAT encoding methods
SAT encoding methods
Direct Encoding (Walsh 2000)
uses a different Boolean variable p(x, a) representing x = a for
each integer variable x and integer value a.
p(x, a)
⇐⇒
x =a
Order Encoding (Tamura et al. 2006)
uses a different Boolean variable p(x, a) representing x ≤ a for
each integer variable x and integer value a.
p(x, a)
⇐⇒
x ≤a
Direct encoding has been widely used in many applications
including GCP solvers, but use of order encoding for
GCP/B-GCP has not been studied.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Order encoding of GCP/B-GCP
In this talk, we propose a new SAT encoding of GCP/B-GCP
based on the order encoding.
Encoding of a vertex vi , that is,
encoding of a variable xi ∈ {1, 2, 3, . . . , k}.
Encoding of an edge {vi , vj } with weight c, that is,
encoding of a constraint |xi − xj | ≥ c (note: c = 1 for GCP).
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Order encoding of a variable xi ∈ {1, 2, 3, . . . , k}
Variable xi ∈ {1, 2, 3, . . . , k} is encoded into the following
Boolean variables and clauses.
Boolean variables
p(xi , 1)
p(xi , 2)
p(xi , 3)
...
p(xi , k − 1)
Clauses
¬p(xi , 1) ∨ p(xi , 2)
¬p(xi , 2) ∨ p(xi , 3)
¬p(xi , 3) ∨ p(xi , 4)
.......
¬p(xi , k − 2) ∨ p(xi , k − 1)
Indira Bhusal Pandey
(if (xi ≤ 1), then (xi ≤ 2))
(if (xi ≤ 2), then (xi ≤ 3))
(if (xi ≤ 3), then (xi ≤ 4))
(if (xi ≤ k − 2), then (xi ≤ k − 1))
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Order encoding of a constraint |xi − xj | ≥ c
The following is an example of encoding x1 ≤ x2 − 3 (x1 , x2 ∈ {1, 2, 3, . . . , 6}).
(x1 ≤ x2 − 3)
→
(x1 ≤ 3) ∧ ¬(x2 ≤ 3) ∧ ((x1 ≤ 1) ∨ ¬(x2 ≤ 4)) ∧ ((x1 ≤ 2) ∨ ¬(x2 ≤ 5))
↔
¬(x1 ≥ 4) ∧ ¬(x2 ≤ 3) ∧ ¬((x1 ≥ 2) ∧ (x2 ≤ 4)) ∧ ¬((x1 ≥ 3) ∧ (x2 ≤ 5))
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Order encoding of a constraint |xi − xj | ≥ c (cont.)
The following is an example of encoding |x1 − x2 | ≥ 3 (x1 , x2 ∈ {1, 2, 3, . . . , 6}).
|x1 − x2 | ≥ 3
→
(x1 ≤ x2 − 3) ∨ (x2 ≤ x1 − 3)
→
(p ∨ q) ∧ (¬p ∨ (x1 ≤ x2 − 3)) ∧ (¬q ∨ (x2 ≤ x1 − 3))
→
(p ∨ q)
∧(¬p ∨ (x1 ≤ 3))
∧(¬p ∨ ¬(x2 ≤ 3))
∧(¬p ∨ (x1 ≤ 1) ∨ ¬(x2 ≤ 4))
∧(¬p ∨ (x1 ≤ 2) ∨ ¬(x2 ≤ 5))
∧(¬q ∨ (x2 ≤ 3))
∧(¬q ∨ ¬(x1 ≤ 3))
∧(¬q ∨ (x2 ≤ 1) ∨ ¬(x1 ≤ 4))
∧(¬q ∨ (x2 ≤ 2) ∨ ¬(x1 ≤ 5))
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Order encoding of a constraint |xi − xj | ≥ c
Step1 : Converting |xi − xj | ≥ c in ≤ form
(xi ≤ xj − c) ∨ (xj ≤ xi − c)
Step2 : Tseitin Transformation
(p ∨ q) ∧ (¬p ∨ (xi ≤ xj − c)) ∧ (¬q ∨ (xj ≤ xi − c))
where p and q are new Boolean variables.
Step3 : Order Encoding
(p ∨ q)
∧
(¬p ∨ (xi ≤ k − c)) ∧ (¬p ∨ ¬(xj ≤ c))
∧
k−c
^
(¬p ∨ (xi ≤ b − 1) ∨ ¬(xj ≤ c + b − 1))
b=2
∧
(¬q ∨ (xj ≤ k − c)) ∧ (¬q ∨ ¬(xi ≤ c))
∧
k−c
^
(¬q ∨ (xj ≤ b − 1) ∨ ¬(xi ≤ c + b − 1))
b=2
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Order Encoding
Developed GCP/B-GCP solver
We developed a GCP/B-GCP solver based on the proposed
method (and direct encoding for comparison purpose).
MiniSat solver 2.0 is used as a SAT solver.
The minimum chromatic number is searched by changing the
chromatic number k with binary search method.
For each of those k’s, GCP/B-GCP is encoded into a SAT
problem, and solved by the MiniSat solver.
In that sense, the developed solver is a complete search solver
which can determine both the lower and upper bounds of the
chromatic number while an incomplete search solver can
determine only the upper bound.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
B-GCP
Performance evaluation of the developed solver
Performance of developed GCP/B-GCP solver (both order
encoding and direct encoding) is compared with previous
works.
Three complete solvers and five incomplete solvers for GCP
and
Three incomplete solvers for B-GCP.
63 GCP and 33 B-GCP benchmark instances of COLOR02
symposium are chosen.
Apple PowerPC G4 with 1.67GHz CPU, 2GB memory is used.
Timeout is set to 30 minutes.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
B-GCP
Number of solved and best results for GCP
Order encoding
Direct encoding
CD (Caramia and Dell’Olmo, 2002)
AVG (van Gelder, 2002)
MZ (Mendez Diaz and Zabala, 2002)
CLGA (Croitoru et al., 2002)
GH2 (Galinier et al., 2002)
BP (Bui and Patel, 2002)
PS (Phan and Skiena, 2002)
CS (Chiarandini and Stuetzle, 2002)
#Solved
21
17
18
4
12
0
0
0
0
0
#Best
34
30
17
5
25
19
10
18
3
13
“#Solved” shows the number of instances for which the optimum
chromatic number is determined by the solver.
“#Best” shows the number of instances for which the best
chromatic number is found by the solver.
Order encoding solver found the optimum for 21 instances and the
best for 34 instances out of 63.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
B-GCP
Number of solved and best results for B-GCP
Order encoding
Direct encoding
PS (Phan and Skiena, 2002)
P02 (Prestwich, 2002)
P08 (Prestwich, 2008)
#Solved
25
15
0
0
0
#Best
26
17
5
22
23
Order encoding solver found the optimum for 25 instances and the
best for 26 instances out of 33.
Order encoding solver shows the best performance compared with
direct encoding solver and three previous solvers.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Example O.E Theorem of O.E
Conclusion
We proposed a new SAT encoding method for GCP/B-GCP
based on order encoding.
We developed a GCP/B-GCP solver using the proposed
method.
The solver is a complete search solver which can determine
both the lower and upper bounds of the chromatic number.
The developed solver shows the best performance compared
with the previous solvers.
The developed solver found the
the best for 34 instances out of
The developed solver found the
the best for 26 instances out of
Indira Bhusal Pandey
optimum for 21 instances and
63 GCP instances.
optimum for 25 instances and
33 B-GCP instances.
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Example O.E Theorem of O.E
Summary of the newly obtained B-GCP results
Improved Benchmark Instances
GEOM60b
GEOM70a
GEOM70b
GEOM80b
GEOM90a
GEOM90b
GEOM120
Prev.
UB
43
62
48
61
64
72
60
Order encoding
LB
UB
41
41
61
61
47
47
60
60
63
63
69
70
59
59
Out of 33 benchmark instances, order encoding solver
improved the upper bound of the chromatic number for the
above 7 instances, and decided the optimum for 6 instances.
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Example O.E Theorem of O.E
Example of Order encoding
1
.
Encoding variables
x, y ∈ {1, 2, 3, 4, 5, 6}
Boolean variables
px, 1 px2 px3 px4 px5
py 1 py 2 py 3 py 4 py 5
Axiom clauses
¬px2 ∨ px3
¬px3 ∨ px4
¬px4 ∨ px5
(similar clauses for y )
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach
Overview GCP/B-GCP SAT Encoding GCP Evaluation Conclusion
Example O.E Theorem of O.E
Theorem of order encoding
Encoding of constraint
∧k−cy ≤ x − c(wherex, y ∈
1, 2, 3, . . . , k) ⇐⇒ b=2 ((¬(y ≤ b − 1) ∨ ¬(x ≤ c + b − 1))
For example, in y ≤ x − 3(x, y ∈ 1, 2, 3, . . . , 6)
(y ≥ 1) ∧ (x ≤ 3) . . . (y ≥ 4) ∧ (x ≤ 6)) violate the constraint
←→ ¬(y ≥ 1) ∧ (x ≤ 3) ∧ ¬(y ≥ 4) ∧ (x ≤ 6) ←→ (y ≤
0) ∧ ¬(x ≤ 3) ∨ . . . ∨ . . .
Indira Bhusal Pandey
Study of Graph Coloring Problems using SAT-based approach