1. Michael Codish, Michael Frank, Amit Metodi
Logic Programming with Max-Clique and its Application to Graph Coloring
morad muslimany
Ben-Gurion University of the Negev
Joint work with:
Michael Codish, Michael Frank, Amit Metodi

2. Logic Programming with Max-Clique and its Application to Graph Coloring
We will present pl-cliquer, a Prolog interface to the cliquer tool for the maximum clique problem.
Agenda:
Problem definitions and examples
pl-cliquer
Usage examples
Experiments

3. Cliques, Maximum Cliques
A clique in a graph 𝐺=(𝑉,𝐸) is a set 𝐶⊆𝑉 of pairwise adjacent vertices (∀ 𝑣 1 , 𝑣 2 ∈𝐶: 𝑣 1 , 𝑣 2 ∈𝐸)
The maximum clique problem is about finding the largest clique in the graph

4. Cliques, Maximum Cliques
A clique in a graph 𝐺=(𝑉,𝐸) is a set 𝐶⊆𝑉 of pairwise adjacent vertices (∀ 𝑣 1 , 𝑣 2 ∈𝐶: 𝑣 1 , 𝑣 2 ∈𝐸)
The maximum clique problem is about finding the largest clique in the graph

5. Graph Coloring
The problem of graph coloring is to find a labeling 𝑐:𝑉→𝑁 of the vertices of the graph such that 𝑖,𝑗 ∈𝐸⇔𝑐 𝑖 ≠𝑐(𝑗).
The minimal graph coloring problem is to color the graph while using as few colors as possible. This number is called the Chromatic Number of the graph.

6. Graph Coloring
The problem of graph coloring is to find a labeling 𝑐:𝑉→𝑁 of the vertices of the graph such that 𝑖,𝑗 ∈𝐸⇔𝑐 𝑖 ≠𝑐(𝑗).
The minimal graph coloring problem is to color the graph while using as few colors as possible. This number is called the Chromatic Number of the graph.

7. Cliques and colorings
Finding cliques is tremendously useful in the problem of Graph Coloring
It is easy to show that the chromatic number of a graph 𝐺, 𝜒(𝐺), is greater than or equal to the size of the maximum clique, as all clique vertices must get different colors.

8. Hardness of these problems
The clique decision problem (is there a clique of size ≤𝑘 in a graph) is NP-Complete
Subsequently, the problem of finding a maximum clique is also NP-Hard.
The graph coloring decision problem is NP- Complete
In particular, the minimal graph coloring problem is NP-Hard

9. Hard but solvable in practice…
Although the problem of maximum-clique is NP-Hard, it is solvable in practice in many graphs by various tools
Cliquer is one such tool, which is a set of C routines for finding cliques in a graph

10. Logic Programming with Max-Clique and its Application to Graph Coloring
We will present pl-cliquer, a Prolog interface to the Cliquer tool for the maximum clique problem.
Agenda:
Problem definitions and examples
pl-cliquer
Usage examples
Experiments

11. pl-cliquer
We have written an interface that connects Cliquer with SWI-Prolog, called pl-cliquer
Provides five high-level predicates, and we will focus on the following three:
graph_read_dimacs_file/5
clique_find_single/4
clique_find_n_sols/6

12. graph_read_dimacs_file(FileName,NVert,Weights,Graph,Options)
Converts a graph in the standard DIMACS format to a Prolog representation as a Boolean adjacency matrix.
Running example graph:
FileName (content)
NVert
Weights
Graph
Options
graph_read_dimacs_file(FileName,NVert,Weights,Graph,Options)

13. graph_read_dimacs_file(FileName,NVert,Weights,Graph,Options)
Converts a graph in the standard DIMACS format to a Prolog representation as a Boolean adjacency matrix.
Running example graph: 7
FileName (content)
NVert
Weights
Graph
Options
graph_read_dimacs_file(FileName,NVert,Weights,Graph,Options)

14. clique_find_single(NVert,Graph,Clique,Options)
Provides an interface to the Cliquer routine by the same name. Finds a single clique in the graph. 7
NVert
Graph
Clique
Options
clique_find_single(NVert,Graph,Clique,Options)

15. clique_find_single(NVert,Graph,Clique,Options)
Provides an interface to the Cliquer routine by the same name. Finds a single clique in the graph. 7
NVert
Graph
Clique
Options
clique_find_single(NVert,Graph,Clique,Options)

16. clique_find_n_sols(MaxSols,NVert,Graph,Sols,Total,Options)
Allows finding several cliques in a graph.
min_weight(3)
max_weight(4)
maximal(false) 10 7
MaxSols
NVert
Graph
Sols
Total
Options
clique_find_n_sols(MaxSols,NVert,Graph,Sols,Total,Options)

17. clique_find_n_sols(MaxSols,NVert,Graph,Sols,Total,Options)
Allows finding several cliques in a graph.
min_weight(3)
max_weight(4)
maximal(false) 10 7 6
MaxSols
NVert
Graph
Sols
Total
Options
clique_find_n_sols(MaxSols,NVert,Graph,Sols,Total,Options)

18. Logic Programming with Max-Clique and its Application to Graph Coloring
We will present pl-cliquer, a Prolog interface to the cliquer tool for the maximum clique problem.
Agenda:
Problem definitions and examples
pl-cliquer
Usage examples
Experiments

19. Solving the Graph Coloring Problem
We model the problem using a constraint language (BEE)
The constraints model is then compiled to SAT using the BEE compiler (written in Prolog)
The SAT instance is solved using a SAT solver via the Prolog interface (BEE allows a variety of solvers at its backend)
The satisfied SAT instance assignment is decoded back to a graph coloring

20. Main Predicate We model the problem using a constraint language (BEE)
The constraints model is then compiled to SAT using the BEE compiler (written in Prolog)
The SAT instance is solved using a SAT solver via the Prolog interface (BEE allows a variety of solvers at its backend)
The satisfied SAT instance assignment is decoded back to a graph coloring

21. ?- graphColoring(Graph,5,Coloring)
Example
Can we solve our running example with 5 colors?
?- graphColoring(Graph,5,Coloring)

22. Example
graphColoring(Graph, 5,
Coloring)

23. ?- graphColoring(Graph,5,Coloring)
Example
?- graphColoring(Graph,5,Coloring)

24. pl-cliquer & graph coloring
As well known, the maximum clique vertices can be preassigned different colors

25. pl-cliquer & graph coloring
As well known, the maximum clique vertices can be preassigned different colors

26. pl-cliquer & graph coloring

27. Example – with pl-cliquer
graphColoring(Graph,5,Coloring)

28. Logic Programming with Max-Clique and its Application to Graph Coloring
We will present pl-cliquer, a Prolog interface to the cliquer tool for the maximum clique problem.
Agenda:
Problem definitions and examples
pl-cliquer
Usage examples
Experiments

29. DIMACS Benchmarks – Selected Results
With pl-cliquer and clique symmetry breaks
Without pl-cliquer
Instance
Colors
Cliquer Time
BEE Time
Clauses
Vars
SAT Time
queen9_9 10
0.01
0.06
12026
547
6.91/sat
myciel5 5
1170
195
0.85/unsat
fpsol2.i.1 65
0.02
0/sat(BEE)
school1_nsh 13
26.6
0/unsat(cl)
BEE Time
Clauses
Vars
SAT Time
0.06
24927
891
14.36/sat
0.01
1697
235
109.24/unsat
2.26
17781
1.87/sat
0.67
216343
4711
793.22/unsat

30. Toronto Benchmarks – Selected Results
With pl-cliquer and clique symmetry breaks
Without pl-cliquer
Instance
Colors
Cliquer Time
BEE Time
Clauses
Vars
SAT Time
pur-s-93 31
0.5
2.85
39613
3.36/sat 30
2.77
37849
1.44/unsat
car-s-91 27
0.06
1.93
407155
12672
/sat
car-f-92
0.2
1.15
179749
7634
0.73/sat
BEE Time
Clauses
Vars
SAT Time
14.43
71911
17.6/sat
7.18
70275
>86400
2.5
19897
2.6
713441
15616
4.35/sat

31. Questions?