Constraint programming models and populationbased simulated annealing algorithm for finding graceful and \(\alpha \)labeling of quadratic graphs
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Abstract
In this research, a new mathematical integer programming model is presented for the graph labeling problem of quadratic graphs. The advantages of this model are linearity and the existence of an objective function. Furthermore, two constraint programming models and a metaheuristics algorithm are also developed to generate feasible graceful labeling and \(\alpha \)labeling for special classes of quadratic graphs. Experimental results on large sizes of graphs from the literature show the efficiency of the proposed model and approach.
Keywords
Graph labeling Graceful labeling Mathematical programming Constraint programming Populationbased simulated annealing1 Introduction
In this paper, it is assumed that we have undirected graphs without loops or double connections between vertices. A graceful labeling of a simple graph \(G=(V, E)\) is a one to one mapping \(\Psi \) of a vertex set V(G) into the set \(\{0,1,2,\ldots , n\}\) with the below property.
If defined for any edge \(e=\left\{ {u,v} \right\} \in E\left( G \right) \), the value \(\varPsi ^{*}\left( e\right) =\left {\varPsi \left( u \right) \varPsi \left( V \right) } \right \) then \(\Psi ^{*}\) is a onetoone mapping of the set E(G) onto the set \(\{1, 2{\ldots }{\vert }E{\vert }\}\) [1].
A graph with graceful labeling is called graceful graph.
An \(\alpha \)labeling of a graph \(G=\left( {V,E} \right) \) is a graceful labeling of G with the additional condition that: there exists a number \(\upgamma \) such that, for any \(e\in E\left( G \right) \) with end vertices \(u,v\in V\left( G \right) \), \(\min \left\{ {\varPsi \left( v \right) ,\varPsi \left( v \right) } \right\} \le \gamma <\max \left\{ {\varPsi \left( v \right) ,\varPsi \left( v \right) } \right\} \) [1].
Graceful and \(\alpha \)labeling are the first labeling methods which were introduced in the late 1960s by Rosa [6]. It is obvious that if a graph has an \(\alpha \)labeling, it is also graceful.
The computational complexity of this problem is not known, but some related problems, such as finding harmonious labeling in a graph, are in NPclass, but it is not known to be in Pclass [2]. The problem of finding a graph is graceful or not can be shown mathematically in special classes of graphs but it is a tedious and difficult approach in many cases. A detailed survey in the field of graph labeling and its results can be found in an excellent survey by Gallian [2].
One of these classes is quadratic graphs. Quadratic graphs which are shown by \(Q\left( {m,n} \right) \) or \(mC_n \) are graphs with m. isomorphic 2regular graphs that each component has n vertices [3].
Despite a large number of papers published on the subject of graph labeling in the literature, there are few particular techniques to be used by researchers to label graphs gracefully or by any other types of labeling. Furthermore, most approaches only applied to small size graphs with special structure. The following part of this context will provide an overview of the literature on this subject.
2 Literature review
As mentioned before, many attempts have been reported by researchers to find a graceful labeling in quadratic graphs.
Rosa in [6] proved that if G has an \(\alpha \)labeling and if all vertices of G are of even degrees, then \(\left {E\left( G \right) } \right \equiv 0\left( {{ mod}~4} \right) \) and G is bipartite. In [6] it is also proved that these conditions are also sufficient if G is a cycle.
Abraham and Kotzig in [7] proved that the graph \(C_{4a} \cup C_{4b}\). has an graceful labeling for all \(a,b>0\). One of the results of Abraham and Kotzig should be mentioned here: if G is a 2regular graph on n vertices and n edges which has a graceful labeling \(\Psi \), then there exists exactly one number \(v_x \) (\(0<v_x <n\).) such that \(\varPsi \left( v \right) \ne v_x \) for all \(v\in V\left( G \right) \). This number \(v_x\). is referred as the missing value of the graceful labeling. Abraham and Kotzig in [7, 8] proved that missing value in quadratic graphs \(Q\left( {m,4k} \right) \) is \(v_x =mk~or~3mk\). Also in this research, they proved that if \(v_x =mk\), then \(\gamma =2mk\), else if \(v_x =3mk\). then \(\gamma =2mk1\). Eshghi in his Ph.D. dissertation has shown that \(3C_{4k} \) has \(\alpha \)labeling for \(k > 1\) [1]. One of the results of his research was to introduce the concept of standard labeling of graphs [1]. Lakshmi and Vangipuram in [9] proved that \(Q\left( {4,4k} \right) \) is graceful. Eshghi also proved that \(5C_{4k} \) has an labeling for all \(k\ge 1\) [1]. He and Salarrezaei showed that \(7C_{4k} \). has an \(\alpha \)labeling for all \(k\ge 1\) [5, 10].
As mentioned in the previous section, many papers have been published recently to show the existence of a special type of labeling for a particular class of graphs. Eshghi and Azimi used a mathematical programming models fording feasible \(\alpha \)labeling in different classes of graphs [11, 12].
In [13], Redl presented mathematical and constraint programming models for finding graceful labeling in graphs. Smith and Puget in [14] have attempt to find graceful labeling in new classes of graphs by using constraint programming. Mahmoudzadeh and Eshghi presented an antcolony algorithm for finding graceful labeling and \(\alpha \)labeling for some special cases [15]. Eshghi and Salarrezaei have applied a tabu search algorithm for finding a feasible \(\alpha \)labeling in quadratic graphs [4, 8].
3 Problem statement
 It is clear that quadratic graphs are bipartite, so the vertices of the graph can be divided into two separate sets. If these sets are shown by X and Y, the members of these sets are placed one after each other. By considering an \(\alpha \)labeling of \(Q\left( {m,4k} \right) \), we have \(X\in \left\{ {0,1,2,\ldots ,\gamma } \right\} \) and \(Y\in \left\{ {\gamma +1,\gamma +2,\ldots ,4mk} \right\} \) so

If \(v_x =mk\,;\gamma =2mk\); then \(X\in \left\{ {0,1,2,\ldots ,2mk} \right\} \left\{ {mk} \right\} \) and \(Y\in \left\{ {2mk+1,2mk+2,\ldots ,4mk} \right\} \).

If \(v_x =3mk\,;\gamma =2mk1;\) then \(X\in \left\{ 0,1,2,\ldots ,2mk1 \right\} \). and \(Y\in \left\{ {2mk,2mk+2,\ldots ,4mk} \right\} \left\{ {3mk} \right\} \).


In this research,, without loss of generality, we consider \(v_x =3mk;\) therefore, \(\gamma =2mk1\): Due to the symmetrical structure of the problem, it is easy to show that in a quadratic graphs \(\left( {m,4k} \right) ,k\ge 1\), graceful labeling with \(v_x =mk\) and \(\gamma =2mk\) is equivalent to the graceful labeling with \(v_x =3mk\) and \(\gamma =2mk1\) [10]. By these assumptions, it is only needed to consider one of these cases in our models.
4 Mathematical model
In this part of the research, the proposed mathematical programming model for finding feasible graceful and \(\alpha \)labeling in quadratic graphs are discussed.In this model, decision variables are defined as the labels of the vertices and edges. In Sect. 4.1, decision variables and terminologies will be explained.
4.1 Nomenclatures
Parameters  

m  Number of connected components in \(Q\left( {m,4k} \right) \) 
k  A quarter of the number of vertices in each component in \(Q\left( {m,4k} \right) \) 
Integer nonnegative variables  

The vertex label  
\(x_i \)  \(x\in X\) 
\(y_i \)  \(y\in Y\) 
\(e_i \)  \(e\in E\) 
0–1 variables  

It is equal to 1 iff  
\(\delta x_{1ij} \)  \(x_i >x_j \) 
\(\delta x_{2ij} \)  \(x_i <x_j \) 
\(tx_{ij} \)  \(x_i =x_j \) 
\(\delta y_{1ij} \)  \(y_{i} >y_j \) 
\(\delta y_{2ij} \)  \(y_i <y_j \) 
\(ty_{ij} \)  \(y_i =y_j \) 
\(\delta e_{1ij} \)  \(e_i >e_j \) 
\(\delta e_{2ij} \)  \(e_i <e_j \) 
\(te_{ij} \)  \(e_i =e_j \) 
Vertex labels  \(x_{2\left( {i1} \right) k+1} ,y_{2\left( {i1} \right) k+1} ,x_{2\left( {i1} \right) k+2},\) \(y_{2\left( {i1} \right) k+2} ,\ldots ,x_{2\left( {i1} \right) k+k} ,y_{2\left( {i1} \right) k+k}\) 
Edge labels  \(e_{4\left( {i1} \right) k+1} ,e_{4\left( {i1}\right) k+2} ,\ldots ,e_{4\left( {i1} \right) k+k} \) 
4.2 Mathematical programming formulation for \(\alpha \)labeling of Q(m, 4k)
Note that in the proposed model parameters can be considered as \(\epsilon =1\), \(M_x =2mk\), \(M_y =2mk+1\) and \(M_e =4m\).
The number of constraints and variables in this mathematical model respectively are \(\left( {4mk\left( {4+15mk} \right) +1} \right) \) and \((120m^{2}k^{2}+8mk)\), respectively.
5 Constraint programming model
Finding a graceful and \(\alpha \)labeling in graphs can be considered as a feasibility problem. As it was shown in the literature, constraint programming technique approach is suitable in finding feasible solutions. In this part, two constraint programming models are presented to find a feasible graceful and \(\alpha \)labeling for quadratic graphs.
5.1 The first constraint programming model
5.2 The second constraint programming model
6 Populationbased simulated annealing
Now, we want to solve the model by simulated annealing (SA). Simulated annealing was introduced by Scott Kirikpatrick [17]. At each step, the algorithm considers some neighbors of the current solution and probabilistically decides whether to move the system into a new state that is selected from neighbors of the current solution or to stay in the current solution. These probabilities ultimately lead the system to move into states of lower energy. Typically this step is repeated until the whole system reaches to a state that is good enough as a solution, or until a given computational stop criterion has been reached. According to this description, the strength of SA is in generating intensification. In order to have adequate diversification in the algorithm, the population aspect in the SA algorithm will be added. Consequently, the population based simulated annealing will be attained, which is called PBSA [16].
Reduce temperature function which is proposed is \(T\left( n\right) =\alpha \cdot T\left( {n1} \right) , \quad 0<\alpha <1\) [17].
Solution representation, objective function, neighborhood structure and stop criteria of our model will be described in the following subsections, respectively.
6.1 The solution representation
The performance in metaheuristic algorithms depends significantly on the representation of the solution. In the proposed algorithm, vertex labels in quadratic graphs are represented in an array with the length of 4mk which it’s odd elements must be members of X and the even elements of this array must be members of Y. For instance, Fig. 2 shows the solution for the graph.
6.2 The objective function
The accomplished calculations express that the utilization of probabilities is not pertinent in the objective function. Table 1, exhibits this fact. This table retells this point that the number of possible state for generating median labels is numerous.
Edges probability distribution
Edge label  The number of possible states  Probability of edge labels equal to i 

\(i\in \left\{ {1,\ldots ,mk} \right\} \)  i  \(\frac{i}{4m^{2}k^{2}}\) 
\(i\in \left\{ {mk+1,\ldots ,2mk} \right\} \)  \(i1\)  \(\frac{{i}1}{4m^{2}k^{2}}\) 
\(i\in \left\{ {2mk+1,\ldots ,3mk} \right\} \)  \(4mki\)  \(\frac{4mki}{4m^{2}k^{2}}\) 
\(i\in \left\{ {3mk+1,\ldots ,4mk} \right\} \)  \(4mki+1\)  \(\frac{4mki+1}{4m^{2}k^{2}}\) 
Standard labeling for 4 component of \(mC_4k,m \ge 5\)
Standard labeling for 1st component  

\(X=\left\{ {0,1,2,\ldots ,k1,k+1,k+2,\ldots ,2k1,2k} \right\} \)  
\(Y=\left\{ {4mk2k+1,4mk2k+2,\ldots ,4mk2,4mk1,4mk} \right\} \) 
Standard labeling for 2nd component  

\(X=\left\{ {2mk2k,2mk2k+1,2mk2k+2,\ldots ,2mk1} \right\} \)  
\(Y=\left\{ {2mk,2mk+1,2mk+2,\ldots ,2mk+k1,2mk+k+1,2mk+k+2,\ldots ,2mk+2k2,2mk+2k1,2mk+2k} \right\} \) 
Standard labeling for 3rd component  

\(X=\left\{ {2mk4k1,2mk4k,\ldots ,2mk3k3,2mk3k2,2mk3k,2mk3k+1,\ldots ,2mk2k2,2mk2k1} \right\} \)  
\(Y=\left\{ {2mk+2k+1,2mk+2k+2,\ldots ,2mk+4k2,2mk+4k1,2mk+4k} \right\} \) 
Standard labeling for 4th component  

\(X=\left\{ {2k+1,2k+2,2k+3,\ldots ,4k2,4k1,4k} \right\} \)  
\(Y=\left\{ {4mk4k,4mk4k+1,\ldots ,4mk3k2,4mk3k1,4mk3k+1,4mk3k+2,\ldots ,4mk2k2,4mk2k\!\!1,4mk2k} \right\} \) 
6.3 Neighborhood structure

First policy: select X or Y or both of them. Then generate two numbers \(R_1 \) and \(R_2 \) such that \(0<R_1\;{\text {and}}\; R_2 <2mk\). If X is selected, relocate \(x_{R_1}\) with \(x_{R_2} \) else if Y is chosen, relocate \(y_{R_1}\) with \(y_{R_2} \). If X and Y are selected, substitute \(x_{R_1}\) by \(x_{R_2}\) and substitute \(y_{R_1}\) by \(y_{R_2}\).

Second policy: select X or Yor both of them. Then generate two numbers \(R_1 \) and \(R_2 \) such that \(0<R_1 \;{\text {and}}\; R_2 <2mk\). If X is selected, values between \(x_{R_1} \) and \(x_{R_2}\) are reversed. If Y is selected, values between \(y_{R_1 } \) and \(y_{R_2 } \) are reversed. If X and Y are selected, reverse any value between \(x_{R_1 } \;{\text {and}}\; x_{R_2 } \) and \( y_{R_1 }\, \& \, y_{R_2 } \).

Third policy: generate two random odd numbers \(i^{\prime }_1 ,i^{\prime }_2 :0<i^{\prime }_1 ,i^{\prime }_2 <4mk\) such that \(i^{\prime }_1 <i^{\prime }_2 \) and move \(V_{i^{\prime }_1 +1} \) and \(V_{i^{\prime }_1 +2} \) to \(V_{i^{\prime }_2 +1} \) and \(V_{i^{\prime }_2 +2} \), respectively, or generate two random even numbers \(i^{\prime \prime }_1 ,i^{\prime \prime }_2 :0<i^{\prime \prime }_1 ,i^{\prime \prime }_2 <4mk\) such that \(i^{\prime \prime }_1 <i^{\prime \prime }_2 \) and move \(V_{i^{\prime \prime }_1 +1} \) and \(V_{i^{\prime \prime }_1 +2} \) to \(V_{i^{\prime \prime }_2 +1} \) and \(V_{i^{\prime \prime }_2 +2} \), respectively.
6.4 Stopping criteria
 1.
Finding a feasible labeling.
 2.
Reaching the predetermined maximum number of iterations.
 3.
Reaching the time limit required by the algorithm.
Time of experimental results of an integer mathematical model (s)
m  k  

1  2  3  4  5  6  7  8  9  10  11  
1  0.12  0.18  0.32  0.54  0.64  3.61  4.61  8.74  12  34  75 
2  0.2  1.59  41  210  38  9.9  13  49  84  –  – 
3  No  4  291  547  845  1067  –  –  –  –  – 
4  2  25  273  874  1027  –  –  –  –  –  – 
5  6.48  68  309  794  1074  –  –  –  –  –  – 
6  41  82  412  815  1056  –  –  –  –  –  – 
7  167  312  954  –  –  –  –  –  –  –  – 
8  415  617  711  819  1074  –  –  –  –  –  – 
9  317  1026  –  –  –  –  –  –  –  –  – 
10  612  –  –  –  –  –  –  –  –  –  – 
Time of experimental results of the first constraint model (s)
m  k  

1  2  3  4  5  6  7  8  9  10  11  12  13  14  
1  0.15  0.19  0.27  0.8  3  0.32  0.8  0.65  0.7  0.6  1.1  0.9  0.7  0.68 
2  0.14  0.18  0.22  1.33  1.42  180  –  –  –  –  –  –  –  – 
3  No  0.56  12  28  62  –  –  –  –  –  –  –  –  – 
4  0.28  0.26  0.54  0.47  0.67  –  –  –  –  –  –  –  –  – 
5  0.31  0.38  0.43  0.5  0.51  –  –  –  –  –  –  –  –  – 
6  0.38  0.42  0.53  0.93  0.63  –  –  –  –  –  –  –  –  – 
Time of Experimental results of the second constraint model (s)
m  k  

1  2  3  4  5  6  7  8  9  10  11  12  13  14  
1  0.17  0.29  0.28  0.32  0.23  0.29  0.32  0.38  0.33  0.35  0.35  0.5  0.42  0.51 
2  0.28  0.28  0.26  0.17  0.26  0.36  0.33  0.4  0.57  0.53  0.36  0.7  0.95  1 
3  No  0.23  0.23  0.42  0.48  0.42  0.64  0.62  0.46  0.46  0.8  0.83  0.61  0.77 
4  0.28  0.26  0.54  0.47  0.67  0.6  0.4  0.62  0.93  1  0.79  0.63  0.94  0.94 
5  0.35  0.38  0.43  0.5  0.51  0.57  1.3  0.79  1  1.7  2  1.7  1.8  2 
6  0.7  0.42  0.53  0.93  0.63  0.87  0.79  1.45  2.1  2.5  3  4.45  8.72  14 
7  0.53  0.48  0.54  0.53  0.51  0.6  1.12  1.45  2.75  2  4.84  5.42  3.11  6.18 
8  0.63  0.57  0.48  0.65  0.73  0.87  0.84  0.93  2.97  8.2  9  15.2  17  19 
9  0.68  0.62  0.74  0.85  0.98  1.33  1.58  3.7  4.53  14  31  420  16.65  16.03 
10  70  0.49  0.62  0.98  1.18  0.68  1.25  2.84  23  20  35  54  730  – 
11  10  0.51  2.23  3.37  4.22  9.12  24  37  42  56  78  910  –  – 
7 Computational experiments
The PBSA algorithm has been coded in MATLAB 2015b. The integer mathematical model and constraint programming models are coded in CPLEX 12.6. All computations were carried out on an Intel\(^{\circledR }\)740QM running at 1.73 GHz with up to 4GB of memory. The time limit for all the test instances in CPLEX was set to 1100 s and in MATLAB was 2700 s. The results are illustrated in Tables 2, 3, 4 and 5.
7.1 Improving approach
As mentioned in Sect. 2, Eshghi in [1] introduced standard labeling for quadratic graphs. With this concept, the investigated problems can be simplified and extended. It is assumed that 4 components of \(Q\left( {m,4k} \right) :m\ge 6\;{\text {and}}\; k\ge 3\) have standard labelings which are illustrated in Table 2, so if corresponding decision variables are fixed, then the algorithm and models must find labeling for the rest (\(m4\)) components of \(Q\left( {m,4k} \right) \).
For improving the efficiency of constraint programming models, a branching rule has been used. This rule is chosen by testing all possible scenarios. Since the search is carried out in the space that is formed with variables x, y and l, then the variables x, y, and l are considered as independent variables and e as dependent.

Branching rule on x variables: from the smallest to the largest indices, variables are selected for generating branches. Then, the branch that is assigned the least value will be selected.

Then, the above procedure will be repeated for y and l variables, respectively.
7.2 Setting parameters
Running the proposed algorithm with the different parameters, but under the same circumstances showed that the best parameters of SA would be set to \(T_0 =100\), \(\alpha =0.95\) . The maximum iteration in interior and exterior loops depended on the graph size as describes in Table 6. Whenever random selection needed in algorithm, a roulette wheel approach used [18].
7.3 Numerical results
Time of experimental results of simulated annealing algorithm (s)
k  m  

1  2  3  4  5  6  7  8  9  10  11  
1  0.52  0.8  No  0.9  1.2  1.35  1.26  2.85  3.25  6.25  7.87 
2  0.47  0.68  0.82  0.95  2.15  3.25  3.12  4.23  6.63  17.23  19 
3  0.64  0.8  1.1  3.2  3.24  4.13  4.25  13.5  18  26  25 
4  0.8  0.37  0.94  4.15  4.29  5.52  6.25  10  11  12.2  37 
5  0.6  0.85  1  9.14  6.42  10.5  14  18  17  41  53 
6  0.29  0.9  0.9  7.13  9.11  7.23  18  25  38  187  125 
7  0.37  0.92  7  10  12  15  20  30  42  215  314 
8  0.44  1.2  0.8  12  16.5  24  22  38  51  328  426 
9  0.35  0.9  1.1  15  18  20  23  42  62  413  542 
10  0.56  1.06  0.45  13  28  38  24  48  73  2325  723 
11  0.71  3.7  2.1  16  25  42  51  62  97  543  685 
12  0.74  0.82  1.17  12.5  27  53  62  81  108  948  1080 
13  0.9  0.8  0.7  9.6  29  13  53  95  105  812  752 
14  1  0.8  0.72  11  32  61  57  89  115  1910  1845 
15  0.98  0.95  0.95  12  61  75  92  112  138  952  2025 
16  1  1.24  8  19  48  82  97  310  423  1097  2214 
17  0.63  1.7  14  24  53  93  107  415  535  1325  2025 
18  1.1  2.11  18  31  67  65  105  416  576  1467  2460 
19  2.21  3.1  23  37  82  83  113  572  750  1724  1152 
20  1.1  6.52  29  42  93  74  128  590  819  1894  2712 
21  0.95  9.27  17  41  104  112  138  617  1121  2305  2510 
22  0.8  23  42  56  531  119  325  812  1960  2101  1142 
23  0.85  42  58  114  286  138  428  923  1655  2411  2674 
24  1.02  45  68  72  394  149  508  1070  2104  2684  – 
25  1.56  48  81  95  423  153  519  1250  1940  2710  – 
Number of iteration in PBSA
Value of m or k  Number of maximum iteration  Number of maximum subiteration 

\( 1\le m\le 2\,{\text {and}}\, 1\le k\le 25~and~3\le m\le 11 \& \, 1\le k\le 2\)  200  50 
\(3\le m\le 7\,{\text {and}}\, 3\le k\le 15\)  500  70 
\(8\le m\le 11\,{\text {and}}\, 3\le k\le 15\)  1000  100 
\( 3\le m\le 11\,{\text {and}}\, 16\le k\le 21~and~ 3\le m\le 6\, \& \, 22\le k\le 25\)  2000  150 
\(7\le m\le 11\,{\text {and}}\, 22\le k\le 25\)  5000  200 
Numerical results in Table 3 show that the mathematical integer programming is able to label quadratic graphs with 160 vertices in a reasonable time. Numerical results in Tables 4 and 5 show that the constraint programming model is able to find a feasible \(\alpha \)labeling for quadratic graphs with 526 vertices very fast.
Numerical results illustrate that the PBSA algorithm is able to labels quadratic graphs with 1012 vertices in a reasonable time.
Figure 3 shows that using the first constraint programming model, we are able to reach a feasible solution in the most cases of small size of quadratic graphs (\(m\le 7\) and \(m \cdot k\le 9\)).
The results of running our model and algorithm imply the following theorems.
Theorem 1
All quadratic graphs \(Q\left( {m,4k} \right) \) for \(1\le m\le 11\) with less than 1000 vertices are graceful and have \(\alpha \)labeling with the exception of \(3C_4 \).
Theorem 2
\(Q\left( {12,4k} \right) \) is graceful and has \(\alpha \)labeling for \(1\le k\le 19\).
Theorem 3
\(Q\left( {13,4k} \right) \) is graceful and has \(\alpha \)labeling for \(1\le k\le 13\).
The details of graceful labeling and \(\alpha \)labeling of the above class of graphs are appeared in the appendix.
8 Conclusion
In this paper, one mathematical integer programming model and two constraint programming models were presented for finding \(\alpha \)labeling and graceful labeling of quadratic graphs. According to our experimental results, for the small size quadratic graphs, both graceful labeling and \(\alpha \)labeling can be found by using the mathematical programming and the first constraint programming model. For the median size quadratic graphs, the second constraint programming was suggested. Furthermore, a populationbased simulated annealing algorithm was also developed to find a feasible solution for large size quadratic graphs.
For future study of this problem, design a hybrid algorithm using PBSA and constraint programming is suggested in this type of graphs.
Supplementary material
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