The Graph k-Colorability Problem (GCP) is a well known NP-hard problem which consist in finding the k minimum number of colors to paint the vertices of a graph in such a way that any two vertices joined by an edge have always different colors. Many years ago, Simulated Annealing (SA) was used for graph coloring task obtaining good results; however SA is not a complete algorithm and it not always gets the optimal solution. In this paper GCP is transformed into the Satisfiability Problem and then it is solved using a algorithm that uses the Threshold Accepting algorithm (a variant of SA) and the Davis & Putnam algorithm. The new algorithm is a complete one and so it gets better quality that the classical simulated annealing algorithm.
Let G=(V,E) be a graph where V is a set of vertices and E is a set of edges. A
kcoloring of G is a partition of V into k sets {V1, …, Vk}, such that no two vertices in
the same set are adjacent, i.e., if v, w belong to Vi, 1 i k, then (v, w) not belong to
E. The sets {V1, …, Vk} are referred to as colors. The chromatic number, x(G), is
defined as the minimum k for which G is k-colorable. The Graph k-Colorability
Problem (GCP) can be stated as follows. Given a graph G, find x(G) and the
corresponding coloring. GCP is a NP-hard problem [
GCP is very important because it has many applications; some of them are
planning and scheduling problems [
The approach used in this paper is to transform GCP into a Satisfiability Problem
(or SAT problem) [
Simulated annealing (SA) [
A deteriorating random move from solution Si to Sj is accepted with a probability exp-(f(Sj)-f(Si))/c. If this move is not deteriorating (the new solution Sj is better than the previous one Si) then it is accepted and a new random move is proposed again. When the temperature is high, a bad move can be accepted. As c tends to zero, SA becomes more demanding through accept just better moves. The algorithm for minimization is shown below:
INITIALIZE(Si=initial_solution, c=initial_temperature) k = 0
Sj = PERTURBATION(Si) If COST(Sj) <= COST(Si) Then
Si = Sj
Si = Sj
The INITIALIZE function starts the initial solution Si and the initial temperature c. The PERTURBATION function makes a random perturbation from Si to generate a neighborhood solution Sj. The COST function gets the cost from a solution. The INC_COST function gets the difference in cost between Sj and Si. Finally, the COOLING function decreases the actual temperature parameter c.
A variant of Simulated Annealing (SA) is the Threshold Accepting method (TA). It
was designed by Dueck & Scheuer [
p(S1, S2) = exp(min{f(S1)-f(S2), 0}/c) COST(Sj) < COST(Si) +T accept Sj (1) (2)
For SAT problems, using a good tune method Threshold Accepting yields better results than Simulated Annealing. This could be because TA does not compute the probabilistic function (1) and does not expend a lot of time making random decisions. The Threshold Accepting algorithm for minimization is the following:
INITIALIZE (Si = initial_solution,
T = initial_threshold or temperature) k = 0
Sj = PERTURBATION(Si) E = COST(Sj) – COST(Si)
Si = Sj
k = k + 1
T = DECREASE_THRESHOLD(T)
Here, the DECREASE_THRESHOLD function is equivalent to the COOLING
function in SA and the threshold T is named “temperature” in order to make more
evident that TA belongs to Simulated Annealing Like Algorithms class (SALA).
SALA uses Simulated-annealing approach with two main loops: internal loop named
Metropolis Cycle and external loop called Temperature Cycle. Number of iterations
in internal and external loop usually are tuned experimentally [
External loop executed from a initial temperature Ti until a final temperature Tf and the internal loop builds a Markov chain of length Lk which depends on the temperature value Tk (k represents the sequence index in Temperature cycle). A strong relation exists between Tk and Lk in a way that:
If Tk , Lk 0 and if Tk 0, Lk .
(3)
Due to TA is applied through a neighborhood structure, V, (PERTURBATION function makes a random perturbation from Si to generate a neighborhood solution Sj), the maximum number of different solutions that can be rejected from Si is the size of its neighborhoods, |VSi|. Then the maximum length of a Markov chain in a TA algorithm is the number of samples that must be taken in order to evaluate an expected fraction of different solutions from VSi at the final temperature Tf, this is:
Lf = C|VSi| . where C varies from 1 C 4.6 (exploration from 63% until 99%), Lf is the length of the Markov chain at Tf.
From (3), Lk must be incremented in a similar but inverse way that Tk is
decremented. Then for the geometric reduction cooling function used by Kirkpartick
[
Tk+1 = Tk.
Lk+1 = Lk.
= exp((ln Lf – ln Li) / n).
Here, Li is the length of the Markov chain at Ti, usually Li = 1, and n is the number of temperature steps from Ti to Tf through (5).
Now, the maximum and minimum cost increment produced through the neighborhood structure are:
ZVmax = Max{COST(Sj) – COST(Si)}.
ZVmin = Min{COST(Sj) – COST(Si)}. (8) (9) (10) (11) for all Sj VSi, and for all Si S
Then Ti and Tf must be calculated as:
Ti = ZVmax .
Tf
ZVmin .
This way of determining the initial temperature enable TA to accept any possible transition at the beginning of the process, since Ti is set as the maximum deterioration in cost that may be produced through the neighborhood structure. Similarly, Tf enables TA to have control of the climbing probability until the algorithm performs a greedy local search.
Satisfiablity Problem [
Let be a propositional formula like formula (12):
F = F1 & F2 & … & Fn (12) where every Fi is a disjunction.
Every Fi is a disjunction of propositional formulas such as X1 v X2 v..v Xr. Every Fi is a clause and every Xj is a literal. Every literal can take a truth value (0 or false, 1 or truth). In Satisfiability problem a set of values for the literals should be found, in such a way that the evaluation of (12) be true; otherwise if (12) is not true, we say that F is unsatisfiable. Besides we say that (12) is in Conjunctive Normal Form or CNF.
The Davis & Putnam method is widely regarded as one of the best deterministic
methods for deciding the satisfiability [
If none rule could be applied, one picks up an arbitrary variable as a branching point and two new formulas are derived by assigning 0 and 1 to this variable. If one of the calls returns with the positive answer the input is satisfiable; otherwise, it is unsatisfiable.
The Davis & Putnam algorithm is shown below:
REDUCE(formula, vreduce)
Return vreduce
valuation=DAVIS-PUTNAM(SUBSTITUTION(true,V, formula)) If valuation != fail Then Return ADD(V=true, vreduce, valuation) valuation=DAVIS-PUTNAM(SUBSTITUTION(false,V, formula)) If valuation != fail Then
Return ADD(V=false, vreduce, valuation)
Else If [C contain V and TF=false]or [C contain ~V and TF=true] Then delete V from C
Return formula
vreduce = empty
formula = SUBSTITUTION(true, V, formula) vreduce = CONS(V=true, vreduce)
formula = SUBSTITUTION (false, V , formula) vreduce = CONS(V=false, vreduce)
The DAVIS-PUTNAM function is the main function and it selects randomly a literal to set a true a group of values in order to create unitary clauses. If that true set values is not the correct solution the complement set of true values is tried. If the new assignment is neither a satisfiable solution, then the formula is unsatisfiable.
The function SUBTITUTION makes the propagation of one literal over all the clauses in formula, deleting clauses where occurs the positive literal L and its value is 1 (true). Therefore the clauses where ~L occurs can delete that literal.
The REDUCE function carries out the search of unitary clauses, so that it can be possible propagate through the function SUBSTITUTION.
Informally coloring a graph with k colors or Graph k-Colorability Problem (GCP) is stated as follows: Is it possible to assign one of k colors to each node of a graph G=(V,E), such that no two adjacent nodes be assigned the same color? If answer is positive we say that the graph is k-colorable and k is the chromatic number x(G). It is possible to transform Graph k-Colorability Problem (GCP) into Satisfiability problem (SAT); that means that for a given graph G=(V,E) and a number k, it is possible to derive a CNF formula F such that F is satisfiable only in the case that G is kcolorable. The formulation of GCP as SAT is made assigning X Boolean variables as follow: 1) Take every node and assign a Boolean variable Xij for every node i and color j; the disjunction of all these variables. In this way every node will have at least one color. Therefore, in the case of figure 1, we have the clauses: 2) To avoid the fact that a node has more that one color, add the formula Xij ~Xik 3) In order to be sure that two nodes (Vi,Vj) connected with an arc have different colors, add a clause such that if Vi has color k, Vj should not be color with this color. This clause is writing as Xik ~Xjk. 4) In order to know which nodes are connected with an edge, an adjacent matrix A of the graph is needed; its elements are:
Aij = 1 0 if i is connected with j otherwise
The reduction of a graph to the Conjunctive Normal Form (CNF) generates so many clauses even for small graphs. For example, for a full graph with 7 nodes (42 edges), 308 clauses with 98 literals can be generated. If we use Davis & Putnam algorithm to color a graph, we could start coloring with R colors (the graph’s degree or from a number given). If it is not possible to color it, then we can increase R and try again.
Due to find a large chromatic number x(G) is a very hard task for a complete method as Davis & Putnam (it demands many resources), we need an incomplete method to help in this task. For this reason we have chosen the Threshold Accepting method. TA will search the chromatic number, but as it is known TA not always get the optimal solution. By this reason, the number found by TA is send to a Davis & Putnam procedure, and this one will get the optimal solution. The complete process is shown in the figure 2.
Any graph can be colored with Gmax+1 colors, where Gmax represents the graph degree. For this reason, TA will try coloring with Gmax colors. If TA gets a success, then TA will try to color with Gmax-1, and so on. When TA finishes, it sends to the output the minimum k of colors founded. In other case, when TA can not color with Gmax colors, then it will send k=Gmax+1 to Davis & Putnam procedure.
Davis & Putnam will attempt to decrease the value of k through binary partitions. The first attempt, Davis & Putnam will choose the number of colors given by (1+k)/2. If the coloring is right, it will color with (1+(1+k)/2)/2 colors, i.e., the left half. Otherwise, the algorithm will color with ((1+k)/2+k)/2 colors, the right half. This process continues until Davis & Putnam cannot decrease k. So, the chromatic number was found. This situation is shown in figure 2.
The figure 3 shows an example where TA found the number nine as its better solution and it is send to Davis & Putnam procedure. When Davis & Putnam takes the last TA solution, using binary partitions and other rules the optimal solution is waited. For example in the case of the figure 3, if Davis & Putnam can not color with five colors, it moves to other alternative, trying with seven colors. Finally, in the last partition, i.e. (7+9)/2, can not color the graph and so the result is a chromatic number equal to nine.
In this paper we presented an algorithm based on Threshold Accepting and Davis & Putnam, to solve the Graph k-Colorability Problem. Because this problem is an NPhard problem there is not a known deterministic efficient (polinomial) method. Nondeterministic methods are in general more efficient but an optimal solution is not guarantee. This method is a new alternative that promises to be more efficient that the previous ones. The main contributions of this paper are enumerated below. 1) We proposed a way to transform the graph k-colorability problem into a satisfiability problem. 2) In order to solve the former problem we proposed a new approach which makes use of the threshold accepting and Davis & Putnam algorithms. 3) The resulting algorithm is complete and using it we can get better results that the wellknown simulated annealing algorithm.