CN114202609A

CN114202609A - Mixed heuristic graph coloring method

Info

Publication number: CN114202609A
Application number: CN202111546250.1A
Authority: CN
Inventors: 吕恒; 周明强
Original assignee: Chongqing University
Current assignee: Chongqing University
Priority date: 2021-12-16
Filing date: 2021-12-16
Publication date: 2022-03-18
Anticipated expiration: 2041-12-16
Also published as: CN114202609B

Abstract

The invention discloses a mixed heuristic graph coloring method, which fuses a probabilistic learning strategy and a mixed evolution algorithm, performs mixed evolution based on two coloring grouping schemes, introduces the probabilistic learning strategy according to the coloring distribution similarity degree of the two coloring grouping schemes before mixed purification, performs coloring dynamic adjustment, enables fast convergence and keeps population diversity to have better balance, further improves the calculation precision while keeping higher graph coloring grouping scheme solving efficiency, and enables the calculation precision and the calculation efficiency of graph coloring problems to be better balanced.

Description

Mixed heuristic graph coloring method

Technical Field

The invention relates to the technical field of graph data processing, in particular to a mixed heuristic graph coloring method.

Background

As an important branch of computer science, graphs (Graph, G) are one of the frameworks in data structures and algorithms. The diagrams may be used to represent virtually any type of structure or system, such as a transportation network, a communications network, circuit board wiring, path planning, playing chess, gaming, optimization processes, task allocation, and human interaction networks, to name a few. In the field of academic or engineering practice, a graph is generally regarded as an abstract network consisting of "vertices" (Vertex), and the vertices in the network can be connected with each other through "edges" (edges) to represent that two vertices are related. Thus, a random, unweighted undirected graph can be generalizedAbbreviated G ═ (V, E), where V ═ V₁,ν₂,...ν_nIs the set of vertices,

is a collection of edges. If there are two vertices e of an edge e ═<v₁,v₂>E, and two vertices v₁And v₂Having the same color, side e being called the conflict side, and adjacent vertex v being called₁And v₂Are conflicting vertices. Further, the graph coloring problem is solved by finding a compliant coloring grouping scheme that avoids conflicting vertices using a minimum number of colors. The minimum number of colors used by the coloring grouping scheme is referred to as the number of colors of the map.

With the recent development of reinforcement learning, the combination of the reinforcement learning and the heuristic algorithm has attracted attention in the academic world. The STAGE algorithm proposed by Boyan and Moore predicts the results of the local search algorithm by learning the evaluation function. By combining a statistical learning strategy and a clustering technology with a memory algorithm, Wang and Tang propose their algorithms to solve the problem of multi-objective flow workshop scheduling. Hutter et al use machine learning techniques (random forest and approximately gaussian processes) to build a predictive model of the algorithm's runtime as a function of the characteristics of a particular problem instance. In 2016, Zhou et al introduced a reinforcement learning based local search (RLS) for solving the grouping problem. The RLS generates an initial solution for a local search algorithm based on descent using a dynamically updated coloring probability matrix. And Zhou et al propose an improved version of a reinforcement learning based local search algorithm (PLSCOL) based on RLS. PLSCOL also uses dynamically updated coloring probability matrices to provide an initial solution for the tabu local search algorithm.

The evolutionary algorithm for solving the problem of graph coloring also has a great deal of research results in the academic world. Population-based evolutionary algorithms are also relatively efficient and popular methods. In 1991, Davis used genetic algorithms to solve the map coloring problem. In 2013, Marappan et al propose single parent conflict gene intersection and conflict gene mutation operators, so that algorithm convergence is faster. On the other hand, as early as 1999, Galinier and Hao embedded the tabu search algorithm into the evolutionary algorithm framework, thereby introducing the Hybrid Evolutionary Algorithm (HEA), obtaining better experimental results, and also providing a brand new idea for the subsequent research. In 2008, Galinier and Hertz propose an Adaptive Memory Algorithm (AMACOL) based on recombination operators, which can obtain the same result as a hybrid evolution algorithm and has the characteristics of simplicity and flexibility. In 2010, Lu and Hao et al proposed adaptive multi-parent crossover operators and distance and quality based pool update replacement criteria, which combined gave good results. For large graphs with high density and many top points, Wu, Hao and the like extract a large independent set by a preprocessing method and then color a residual image by a modular factor algorithm, so that improved results are obtained on 11 graphs with large scale. In 2015, Moalic and Alexandre proposed a Hybrid Evolution Algorithm (HEAD) based on two individuals, and provided population diversity by introducing elite individuals again at a certain number of iterations, which made a great breakthrough in shortening the solution time, and in 2018, Moalic and Alexandre introduced random crossover operators in (HEAD) for improvement.

The hybrid evolution algorithm HEAD proposed by Moalic and Alexandre has a great advantage in solving time, and mainly utilizes a tabu search algorithm and a population of two individuals, so that the algorithm does not need to be a complicated selection replacement strategy. HEAD has the advantages that the population can be converged quickly; the defect is that only two individuals cause insufficient diversity of the population, and the population is difficult to jump out when the population falls into local optimum. Although the HEAD introduces elite individuals in the iterative process, for sparse graphs such as le450_15c and le450_15d, the algorithm easily reaches a convergence threshold value in the iterative process, and the resolving precision is insufficient, so that the obtained graph coloring grouping scheme is not an ideal solution, and thus the impatience is involved.

Therefore, how to find a graph coloring method with better balance between fast convergence and keeping population diversity is a problem to be solved in the field, which gives better consideration to the resolution precision and the resolution efficiency of the graph coloring problem.

Disclosure of Invention

Aiming at the defects in the prior art, the invention provides a mixed heuristic graph coloring method, which is used for further improving the resolving precision while keeping higher solving efficiency of a graph coloring grouping scheme, so that the resolving precision and the resolving efficiency of the graph coloring problem are balanced and considered better.

In order to solve the technical problems, the invention adopts the following technical scheme:

a mixed heuristic graph coloring method comprises the following steps:

s1: randomly generating two initial solutions for the vertex coloring of the graph to obtain two coloring grouping schemes; step S2 is executed;

s2: according to a preset conditional grouping strategy, when the two coloring grouping schemes meet the conditional grouping strategy, color distribution is carried out on each grouping in each coloring grouping scheme according to the corresponding coloring probability matrix, and the vertexes in each grouping are recolored respectively; each coloring probability matrix is used for indicating the probability of color allocation for each group in one coloring grouping scheme; if the condition grouping strategy is not met, keeping the two coloring grouping schemes unchanged; then, step S3 is executed;

s3: performing greedy cross iteration on the two colored grouping schemes by using a greedy cross algorithm, and executing the step S5 when an iteration termination condition is met after the iteration; otherwise, go to step S4;

s4: updating the coloring probability matrix, and then returning to the step S2 for the two coloring grouping schemes after iteration;

s5: and the optimal solution in the two coloring grouping schemes after greedy cross iteration is used as the final vertex coloring grouping scheme of the graph.

Preferably, in step S2, the preset conditional grouping policy specifically includes:

calculating a coloring layout similarity proxi between two coloring grouping schemes:

proxi＝q/n；

wherein q represents the sum of the weights of the maximum weight matching result of the grouping sharing vertex number between the two coloring grouping schemes; n represents the total number of vertices of the graph;

if the coloring layout similarity proxi between the two coloring grouping schemes is greater than or equal to the preset coloring layout similarity threshold proxi₀If yes, judging that the condition grouping strategy is met; otherwise, judging that the condition grouping strategy is not met.

Preferably, the coloring layout similarity threshold proxi₀The value range of (A) is 0.8-0.95.

Preferably, in the greedy cross iteration of step S3, the two coloring grouping schemes are labeled as a first parent and a second parent, respectively; in step S3, the greedy cross iteration method specifically includes:

s301: selecting a group with the largest number of vertexes from the first parent, coloring the group as the first group of the filial parent, and eliminating the corresponding vertexes of the first parent and the second parent; then, let the child determine the number of packets k equal to 1, go to step S302;

s302: selecting a group with the largest number of vertexes from the current second parent, coloring the group as the (k + 1) th group of the filial parent, and removing the corresponding vertexes from the first parent and the second parent; then, judging whether K +1 is equal to K, wherein K is the total number of the groups in the coloring grouping scheme; if yes, executing step S304; if not, enabling the child to determine that the group number k is added by 1, and then executing the step S303;

s303: selecting a group with the largest number of vertexes from the current first parent, coloring the group as the (k + 1) th group of the filial parent, and removing the corresponding vertexes from the first parent and the second parent; then, judging whether K +1 is equal to K; if yes, executing step S304; if not, enabling the child to determine that the number k of the groups is self-added by 1, and then returning to execute the step S302;

s304: judging whether the first parent and the second parent have residual vertexes; if yes, randomly distributing the remaining vertexes to a group of child parents, and executing the step S305; if not, directly executing step S305;

s305: and taking the first parent before the iteration as a new second parent, and taking the current obtained filial parent as a new first parent to finish the greedy cross iteration.

Preferably, in step S301 and step S303, if there are a plurality of groups with the highest number of vertices in the first parent, one group with the highest number of vertices in the first parent is randomly selected from the groups.

Preferably, in step S302, if there are more than one group with the highest number of vertices in the second parent, one group with the highest number of vertices in the second parent is randomly selected from the group.

Preferably, the iteration termination condition is that the number of greedy cross iterations reaches a preset upper limit of the number of iterations, or the obtained new first parent is a compliant coloring grouping scheme.

Preferably, in step S5, the optimal solution of the greedy cross-iterated two coloring grouping schemes is the first parent of the two coloring grouping schemes.

Preferably, in step S4, the method for updating the color probability matrix specifically includes:

for a vertex in a group in the coloring grouping scheme:

if the color grouping of the vertex before and after iteration is changed, updating the coloring probability corresponding to the vertex in the coloring probability matrix according to the following formula according to whether the maximum weight matching strategy is satisfied between the two color groupings of the vertex before and after iteration:

if the color grouping of the vertex before and after iteration does not change, updating the coloring probability corresponding to the vertex in the coloring probability matrix according to the following formula according to whether the color grouping of the vertex before and after iteration meets the maximum weight matching strategy or not:

wherein, P'_ijIs the ith vertex v in the figure_iCorresponding updated coloring probability, representing updated vertex v_iProbability of assignment to jth color; p_ijIs the ith vertex v in the figure_iCorresponding pre-update coloring probability representing pre-update vertex v_iProbability of assignment to jth color; k is the total number of packets in the coloring grouping scheme; alpha, beta and gamma are all coloring probability coefficients, and the values are all larger than 0 and smaller than 1.

Compared with the prior art, the mixed heuristic graph coloring method disclosed by the invention has the advantages that the probabilistic learning strategy and the mixed evolution algorithm are fused, the mixed evolution is carried out based on the two coloring grouping schemes, the probabilistic learning strategy is introduced according to the coloring distribution similarity degree of the two coloring grouping schemes before the mixed purification, the coloring dynamic adjustment is carried out, the fast convergence and the population diversity are better balanced, the higher graph coloring grouping scheme solving efficiency is kept, the calculating precision is further improved, and the better balance and consideration are obtained on the calculating precision and the calculating efficiency of the graph coloring problem.

Drawings

FIG. 1 is a flow chart of a hybrid heuristic graph coloring method of the present invention.

Fig. 2 is a diagram of maximum weight matching results between two coloring grouping schemes and an example of coloring layout similarity calculation.

FIG. 3 is an exemplary diagram of a one-time greedy cross-iteration process for two coloring grouping schemes.

Detailed Description

The invention introduces a dual Hybrid Evolution Algorithm (HEAD) and a local search algorithm (PLSCOL) based on reinforcement learning, which are commonly used for solving the graph coloring problem at present, and have excellent performances on most DIMACS reference examples, and are algorithms compared with the method in the experiment.

1 Dual Mixed evolution Algorithm (HEAD)

The Hybrid Evolution Algorithm (HEA) is an excellent hybrid evolution algorithm combining tabu and GPX, which are taboo search coloring algorithms, and proposed by moralic et al in 2015. Since the tabu search algorithm is a very strong algorithm, it leads to fast convergence in population iterations. GPX provides a diverse population, and the overall convergence speed of the algorithm is slowed down while the original solution structure is not damaged, so that HEA can well balance intensification and diversification, and a better experiment result is obtained. In 2018, Moalic et al further improved HEA, and by introducing elite individual (elite individual), further increased population diversity and obtained better experimental results.

By g_iRepresenting groupings of vertices of color i, a legal K coloring S can also be considered as dividing the set of vertices V into K groups (also called independent sets), S ═ g₁,g₂,...,g_KSuch that adjacent vertices are not in the same group. For two solutions S ═ g₁,g₂,...,g_KAnd S '═ g'₁,g'₂,...,g'_KIf for

Total and only one g'_jE S' such that g_i＝g′_jThat is, there is a one-to-one correspondence between the two groups, S and S' are called to be equivalent, and are recorded as: s ═ S'. In other words, whether one solution is equivalent to another depends only on the relationships between the vertices within the packet, not on the color of the vertices themselves.

The execution of greedy crossing (GPX) is now briefly described. Consider two parents, S₁＝{G₁ ¹,G₂ ¹,...,G_K ¹And S₂＝{G₁ ²,G₂ ²,...,G_K ²Divide the vertices into K groups according to color classes, and simultaneously for each group

Indicating that the vertex V belongs to parent 1 and i represents the color i. First, selecting the group with the largest number of vertices from parent 1, and coloring the corresponding vertices in the children with the color of the group. When the largest packet is not unique (i.e., as many colors as there are packets), one of the packets is randomly selected. While processing parent 1, the corresponding vertices in parent 2 are culled (given that these vertices have been passed on to children through parent 1). It can be seen that parent 2 may therefore lose the largest color grouping, and then proceed to the second step of selecting the largest color grouping from parent 2 and coloring the corresponding vertices in the children. At the same time, the corresponding color in parent 1 is also rejected. At this point, a round of iterations is completed, i.e., vertices are selected from the two parents and their colors are inherited into the children. This process ends with all colors of children being colored. It should be noted that when the iteration proceeds to the last step, there are still some un-colored vertices in the children, which will be randomly assigned colors, and the result is that when the two parents are very different, the last step of GPX will generate many vertices that need to be randomly colored, thus generating many conflicts in the children. This is also the reason why GPX mentioned in the article by the authors moralic appears to be variant. Also, the order of parent selection affects the generation of progeny.

Algorithm 1 is pseudo code for the HEAD algorithm.

Algorithm 1 can be seen as two parallel tabucl algorithms iteratively iterated through a periodic crossover operation. After randomly generating two initial solutions, the algorithm enters an iterative state until a termination condition is reached. Firstly, introducing some diversity for the population through a crossover operator GPX, and then solving two generated filial generations mainly through a TabuCol algorithm. Secondly, by introducing elite individuals to participate in the solution, with the aim of injecting diversity again for the population, the coloring layout similarity arising from the two solutions in the iteration will rise. It can be seen that in an iteration of a cycle, the solution elite1 is the best solution in the current cycle, the solution elite2 is the best solution in the previous cycle, and after a cycle, the elite2 will replace one of the individuals. Indeed, the introduction of elite individuals not only preserves the best solutions, but also reintroduces diversity into the population in iterations.

Local search algorithm based on probability learning (PLSCOL)

The PLSCOL algorithm is a probabilistic learning based local search algorithm proposed by Zhou et al in 2018. In the iterative process of a local search algorithm Tabucol, a probability learning strategy is introduced, namely before each local search, the vertex is enabled to select a corresponding color group through a coloring probability matrix, and the vertex coloring is adjusted globally. And calculating a color matching relation through the change of the vertex color between the solutions, and updating the coloring probability matrix to provide grouping basis for the next iteration.

Algorithm 2 uses the colored probability matrix to generate an initial solution for local search. The generated local optimal solution establishes a matching relation between a group and the grouping in the initial solution, the matching relation is obtained through a maximum weight matching algorithm, and the coloring probability matrix is updated according to the change of the vertex color. It should be noted that for each P_ijIt represents the ith vertex v in the diagram_iThe probability assigned to the jth color. For P_ijOn one hand, the color corresponding relation is found through maximum weight matching, and on the other hand, P is changed through a probability updating strategy_ijThe value of (c). Meanwhile, for each probability update, P is smoothed by a strategy of probability smoothing_ij。

3 Mixed heuristic graph coloring method provided by the invention

On the basis of the hybrid evolutionary algorithm HEAD of Moalic et al and the PLSCOL algorithm of Zhou et al, the invention combines a reinforcement learning strategy with an evolutionary algorithm, and provides a new hybrid heuristic graph coloring method which can be marked as PLHEAD.

3.1 Mixed heuristic graph coloring method PLHEAD

The HEAD proposed by Moalic and Alexandre has a great advantage in solving time, and a tabu search algorithm and a population of two individuals are mainly utilized, so that the algorithm does not need to be a complicated selection and replacement strategy. HEAD has the advantages that the population can be converged quickly; the defect is that only two individuals cause insufficient diversity of the population, and the population is difficult to jump out when the population falls into local optimum. Although the HEAD introduces elite individuals in the iterative process, for sparse graphs such as le450_15c and le450_15d, the algorithm easily reaches a convergence threshold value in the iterative process, and the resolving precision is insufficient, so that the obtained graph coloring grouping scheme is not an ideal solution, and thus the impatience is involved.

The invention integrates a probabilistic learning strategy and a hybrid evolutionary algorithm, finds balance between rapid convergence and population diversity maintenance, and provides a hybrid heuristic graph coloring method PLAHEAD. The algorithm framework pseudo code may be as shown in table 3.

The algorithm 3 takes a hybrid evolution algorithm as a framework, before GPX crossover operation is carried out, a grouping strategy is firstly carried out according to a condition (line 4), after a grouping condition is met, a groupselection algorithm is used for each parent, a color (line 5 and line 6) is selected for each color in each parent according to probability through a coloring probability matrix, the coloring probability matrix is used as an attribute of a solution and is inherited to descendants (line 8 and line 9) along with the GPX crossover operation, and particularly, the coloring probability matrix of which parent is selected depends on a first input parent of the GPX crossover operation. Two children (line 10 and line 11) are obtained after the tabu search algorithm. And continuously updating the coloring in iteration according to the Probability update strategyProbability matrix P_p1And P_p2(line 12 and line 13). The elite individuals are then saved and introduced into the population when a certain iteration period is met (lines 14-19). Although the original solution structure may be damaged by adding such a grouping strategy, the update of the coloring probability matrix has certain randomness, and the structure which is to be converged to the final solution structure may be easily broken, a strategy grouped according to the probability is not introduced in each iteration, but a dynamic adjustment is performed according to the similarity degree of the coloring distribution of two parents, and the original solution structure is less easily broken by the strategy adjusted according to the satisfied condition than the adjustment mode of a fixed period, so that the time loss caused by repeated calculation can be reduced.

Therefore, summarizing the flow of the hybrid heuristic graph coloring method of the present invention is shown in fig. 1, and comprises the following steps:

s2: according to a preset conditional grouping strategy, when the conditional grouping strategy is met, color distribution is carried out on each group in each coloring grouping scheme according to a corresponding coloring probability matrix, and the vertexes in each group are recolored respectively; each coloring probability matrix is used for indicating the probability of color allocation for each group in one coloring grouping scheme; if the condition grouping strategy is not met, keeping the two coloring grouping schemes unchanged; then, step S3 is executed;

3.2 conditional grouping strategy

For each iteration, if a probabilistic grouping strategy is used without limitation, the original gradually-converged solution structure is damaged, so that the efficiency and quality of solution are reduced, corresponding experimental data are given subsequently to explain the situation, and meanwhile, a conditional grouping strategy is provided in view of the above.

There is a relationship between two parents. According to the coloring layout of the two (parent) solutions, the degree of similarity of the coloring layout is adopted as coloring layout similarity (the parameter is denoted as proxi), and the calculation is shown as the formula (1).

proxi＝q/n； (1)

Wherein q represents the sum of the weights of the maximum weight matching result of the number of the grouped sharing vertexes between the two coloring grouping schemes, namely the total number of the two parents for each color sharing vertex; n represents the total number of vertices of the graph.

Example as shown in fig. 2, there are existing coloring grouping schemes S1 and S2 of a graph with 10 vertices (a, B, C, D, E, F, G, H, I, J), and fig. 2(a) gives an example of coloring layout similarity calculation: the known coloration number is 3; two solutions are respectively provided with 3 groups, and the two solutions are subjected to maximum weight matching (wherein the edge weight value represents the intersection of the two groups). At this time obtain

And the sum of the weights is 3+2+1 to 6. Fig. 2(b) abstracts the bipartite graph form.

The sum of the weights of the maximum weight matching results in the above example is 6, and the total number of vertices in the graph is 10, then the coloring layout similarity can be expressed as 6/10 ═ 0.6, i.e., the coloring layout similarity between the two solutions is 60%. It can be seen that although the color layout of each solution is very different, a more reasonable answer can still be obtained by the coloring layout similarity calculation, which is also a key parameter in the method of the present invention. And when the current coloring layout similarity is larger than or equal to the coloring layout similarity threshold value through iteration, the probability grouping condition is met, a grouping selection strategy is started, and changes are brought to the population.

Therefore, it can be summarized that the conditional grouping strategy preset in the hybrid heuristic graph coloring method of the present invention is specifically:

proxi＝q/n；

3.3 greedy Cross-iteration strategy

In the hybrid heuristic graph coloring method, when greedy cross iteration processing is performed on the two colored grouping schemes in step S3, in order to distinguish the two colored grouping schemes, the two colored grouping schemes are respectively marked as a first parent and a second parent; the greedy cross iteration method specifically comprises the following steps:

wherein if there are more than one group with the highest number of vertices in the first parent, one group with the highest number of vertices in the first parent is randomly selected from the group.

wherein if there are more than one group with the highest number of vertices in the second parent, one group with the highest number of vertices in the second parent is randomly selected from the group.

S304: judging whether the first parent and the second parent have residual vertexes; if yes, randomly distributing the remaining vertexes to a group of child parents, and executing the step S305; if not, step S305 is directly executed.

The greedy cross iterative process is to sequentially and alternately extract groups between a first parent and a second parent for coloring until the number of the extracted groups reaches the total number of the groups in the coloring grouping scheme, and if the uncolored vertexes exist, randomly distributing the vertexes to one group of the filial parents for coloring to form a complete sub-band parent.

An example of this greedy cross-iterative process may refer to FIG. 3, with coloring grouping schemes S1 and S2 for a graph with 10 vertices (A, B, C, D, E, F, G, H, I, J), colored color books of 3 colors, red, blue, and green, respectively; the resulting progeny parents are [ (D, E, F, G) red cohort, (B, H, J) blue cohort, (a, C, I) green cohort ].

And the iteration termination condition of the greedy cross iteration process is that the number of the greedy cross iteration reaches a preset iteration number upper limit, or the obtained new first parent is a compliant coloring grouping scheme.

Through the iteration process, some diversity is introduced into the population through the crossover operator GPX, the population is solved by introducing elite individuals, the purpose is to inject diversity into the population again, and the coloring layout similarity derived from two solutions in iteration is increased. Meanwhile, after each iteration, the obtained new first parent is the best coloring grouping scheme solution in the current iteration cycle, and the new second parent is the best coloring grouping scheme solution in the previous iteration cycle. The introduction of elite individuals not only preserves the best solution, but also reintroduces diversity into the population in iterations.

Therefore, if the greedy cross-iteration process is terminated and the optimal solution of the two colored grouping schemes after the greedy cross-iteration in step S5 is skipped, the optimal solution is the first parent of the two colored grouping schemes.

3.4 coloring probability matrix update strategy

After each iteration, the color probability matrix is updated in step S4. As the simplest update scheme, the coloring probability matrix can be updated in a random update manner. But in order to better satisfy the principle of maximum weight matching between parents and children of an iteration, the scheme of the present invention preferably updates the color probability matrix in the manner of the following algorithm 4.

In the update scheme, two specific modes are mainly used for updating the color probability matrix:

for a vertex in a group in the coloring grouping scheme:

the first mode is as follows: if the color grouping of the vertex before and after iteration is changed, updating the coloring probability corresponding to the vertex in the coloring probability matrix according to the following formula according to whether the maximum weight matching strategy is satisfied between the two color groupings of the vertex before and after iteration:

and a second mode: if the color grouping of the vertex before and after iteration does not change, updating the coloring probability corresponding to the vertex in the coloring probability matrix according to the following formula according to whether the color grouping of the vertex before and after iteration meets the maximum weight matching strategy or not:

wherein, P_ij' is the ith vertex v in the figure_iCorresponding updated coloring probability, representing updated vertex v_iProbability of assignment to jth color; p_ijIs the ith vertex v in the figure_iCorresponding pre-update coloring probability representing pre-update vertex v_iProbability of assignment to jth color; k is the total number of packets in the coloring grouping scheme; alpha, beta and gamma are all coloring probability coefficients, and the values are all larger than 0 and smaller than 1.

When the coloring probability matrix is updated to a certain degree, the probability value is out of bounds. For example, when a certain probability value is continuously increased in iteration, even exceeds 1, and the legal value of the probability is [0,1], certain correction needs to be made on the illegal probability value, the probability distribution is adjusted for the whole packet with the probability value exceeding 1, and the persistence and the correctness of the algorithm are ensured.

3.4 added overhead compared to HEAD Algorithm

It is known from the algorithm that after each iteration of the local search algorithm, a probability update is accompanied, and the probability update algorithm needs to be applied to each vertex V_iThe grouping judgment is performed so that V is determined for each vertex_iAll are accompanied by P_ijIn which j ∈ [1, K)]Can obtain time complexThe degree of impurity is O (n.times.K). On the other hand, the updated source of the coloring probability matrix is based on the result of the maximum weight matching between parent and child, and the time complexity of the maximum weight matching algorithm is O (n)³) Therefore, each probability update is obtained along with one maximum weight matching algorithm and one P matrix update, and the total time complexity is O (n)³) + O (n × K), is a polynomial level algorithm.

On the other hand, when the similarity of the coloring layout of the two parents reaches the threshold, GroupSelecting is performed, that is, each vertex V is subjected to_iAccording to P_ijThe probability value of the group is selected to belong to the group, one part of the vertexes is selected to be the group with the maximum probability value, and the other part of the vertexes is selected randomly, so that the similarity of the coloring layout of the two originally converged parents is reduced, and the phenomenon that the cross strategy is stopped due to the loss of action is avoided. This process, each vertex V_iThe coloring judgment is performed so that each vertex V is_iAll accompanied by selection of P_ijOr randomly selected, where j ∈ [1, K ]]The time complexity is O (n × K).

4. Comparative test and results

First, the selected data set is described, and the parameter settings and meanings of the experiment, and the detailed result display and analysis are described in detail. Comparison with the best results at present will also demonstrate the improved effectiveness of the method of the invention.

4.1 data set introduction

The data set for the methodological test contrast experiment was primarily the second DIMACS challenge baseline data set, where most of the plots were random or quasi-random. A total of 34 datasets, a total of 6 categories of datasets, a first category of 12 random graphs, vertices randomly distributed in unit squares, and their coloring numbers unknown (dsjc125.x, dsjc250.x, dsjc500.x, dsjc1000.x, wherein 125,250,500,1000 respectively represent the number of vertices, and x respectively takes values of 1,5, and 9 respectively represent graphs having densities of 0.1,0.5, and 0.9, and the definition of density d is related to the number of vertices and the number of edges (d ═ 2m/n (n-1), wherein m is the number of edges, and n is the number of vertices, the denser the graph is represented, and the sparser the second category is 12 leighton (leighton) graphs, and their coloring numbers are known (le450_5x, 450_15x, 450_25x, wherein 450 is the number of vertices of graphs, 5,15 x, 25 a, and 4 d is represented by a plane, respectively, and the third category is 4 d plane graph, their numbers of coloring are also known (flat 300-28-0, flat 1000-x-0, x: 50, 60, 76, respectively, representing the number of colors in the graph, where 300, 1000 are the number of vertices in the graph), the fourth is the geometric random graph dsjr500.1c, the vertices 500 are randomly distributed in unit squares, and the density is 0.1. The suffix c represents the complement, so the edge density of this example is 0.9, and the fifth class is also the geometric random graph (r250.5, r1000.1c, r 1000.5). The sixth category is two large random graphs, c2000.5 and c 4000.5. Where 2000 and 4000 represent the number of vertices, their edge density is 0.5. The main reason for choosing these figures is that they are extensively studied in the literature and are relatively difficult figures, and it is only with such comparative experiments that are meaningful. The details of the data set are shown in Table 1, where the example name, number of vertices, number of edges, edge density, and known number of colorations of the data set are listed, respectively.

Table 1 example of data sets

4.2 results of the experiment

The PLHEAD algorithm is realized by C + +, is compiled by GNU g + +, and is an experiment environment of a Linux 64-bit system with a processor Intel Xeon (Skylake) Platinum 81632.5 GHz and a memory of 4 GB. For most examples, the number of solutions set is 20. The end condition of each solution is that a single operation reaches the upper limit of 2000 iterations, or a legal K coloring grouping scheme is found. For the comparison algorithm HEAD, the same experimental environment was used for the comparison experiment, and the results of the comparison experiment are given in Table 2.

TABLE 2 comparison of experimental results of the PLAHEAD method of the present invention and the HEAD algorithm

First, a comparison of experimental results of PLHEAD versus HEAD is presented. The purpose of this set of experiments was to compare the performance of the plan algorithm based on two strategies with the original plan algorithm. Columns 1-2 of Table 2 give the example name and the number of colors (K) set. For different examples, the number of iterations of tabu search set in the two algorithms is different (iterTc), and for the two algorithms, a running success rate (success rate), an average number of iterations (Iter), an average number of crossings (Cross), an average running time (times (s)) and a conditional grouping threshold (proxi) for PLHEAD are given separately₀). The choice of threshold is explained in table 7. For the experimental results of the same example, the party with the predominance in time or the predominance in success rate is bolded, and the predominance of the number of iterations is underlined.

Table 2 indicates that the performance of the plan is superior to the plan in most cases, where the success rate improvement for the two data sets of le450_15c and le450_15d is significant, being improved by 95% and 90%, respectively. Of all 37, there were 19 instances or better than HEAD in terms of time, success rate and number of iterations. The experiments prove that the plan conditional grouping and probability updating strategy is improved to a certain extent in solving the graph coloring problem compared with the original plan.

TABLE 3 Complicson of experimental results between PLHEAD and PLSCOL

Table 3 shows the results of a comparison experiment between PLHEAD and PLSCOL. The first column of table 3 gives the name of the instance. . For different examples, different coloring numbers (K) are set in the two algorithms, and the success rate of operation (# success), average number of iterations (# Iter), and average running time (times (s)) are given, respectively. Similarly, for the experimental results of the same example, the party with the predominance in time or the predominance in success rate is highlighted with bold letters, and the predominance in iteration times is underlined.

Table 3 indicates that PLHEAD performs significantly better than PLSCOL in most of the examples, and that PLHEAD performs significantly better than PLSCOL in the 20 examples given, with 11 examples performing better than PLSCOL individually or in terms of time, number of tints, number of iterations, and success rate of runs. Wherein the coloring number of the 4 instances dsjr500.5, flat1000_76_0, r250.5 and r1000.5 is less than that of PLSCL. The runtime of the PLHEAD at 7 instances is significantly dominant, with a time improvement of dsjc500.5 of more than 20 times, a solution time improvement of djsc1000.9 of more than 4 times, and improvements of le450_15c and le450_15d of more than 20 times. This set of experiments demonstrates that PLHEAD has certain advantages over PLSCOL in solving the graph coloring problem based on the conditional grouping and probability updating strategy of the hybrid evolutionary algorithm framework.

TABLE 4 Comprison of experimental results between the PLAAD and the PLAAD with out updating

Table 4 is a comparison experiment of PLHEAD and PLHEAD disabling update strategy. The purpose of this set of experiments was to demonstrate the effectiveness of the update strategy in solving the whole of the PLHEAD. As can be seen from table 4, the performance of the plan under the update-disabled strategy is not as stable as the original plan. Results of experiments with 17 total examples show excellent results of PLHEAD in time, number of iterations or success rateIn versions where updates are disabled, the temporal complexity is O (n) due to the update policy⁸) Such that the plan is not as fast as disabling the updated version in solving for the simple instance. This set of experiments demonstrated the effectiveness of the renewal strategy for PLHEAD as a whole.

TABLE 5 Comprison of experimental results between the PLAAD and the PLAAD with out promoting selection

Table 5 is a comparative experiment of PLHEAD versus PLHEAD disable conditional grouping strategy. For each iteration, if a probabilistic grouping strategy is used without limitation, the original gradually converged solution structure is damaged, so that the efficiency and quality of the solution are reduced. The purpose of this set of experiments was to demonstrate the effectiveness of the condition grouping to improve overall solution for PLHEAD.

As can be seen from table 5, the performance of the plan under the disabling conditional grouping strategy is much less stable than the original plan. The results of the experiments for a total of 23 examples outperformed the version of the forbidden conditional packet in time, number of iterations, or success rate. For some examples, an algorithm that disables the conditional grouping strategy cannot complete the solution with the same limit on the number of tints K. The set of experiments proves the importance of the conditional grouping strategy to the PLHEAD as a whole, and simultaneously can ensure the stability of the whole iteration.

TABLE 6 comprehensive of experimental results between the PLAAD and the PLAAD with out promoting selecting and updating

Table 6 is a comparative experiment where PLHEAD and PLHEAD disable both the conditional grouping strategy and the update strategy. The purpose of the set of experiments is to prove the necessity of improving the overall solution of the PLHEAD under the combined action of the condition grouping strategy and the updating strategy.

As can be seen from table 6, the performance of the plan under simultaneous disabling of both strategies is much less stable than the original plan. The results of the experiments for a total of 23 cases outperformed the version with all policies disabled in time, number of iterations or success rate. For some examples, the algorithm that disables the strategy cannot complete the solution with the same limit on the number of tints K. For part of simple graphs, the iteration times are limited, and a legal solution is obtained after one or two rounds of tabu search are completed, the improvement on the solution efficiency caused by forbidding the updating strategy is large, so that the solution time of the examples is dominant, and on the whole, the group of experiments prove that the condition grouping strategy and the updating strategy are significant for improving the PLHEAD.

TABLE 7 proxi of antibiotics in DIMACS

A regroupselecting threshold is set for each instance based on the density of each instance as a reference. Table 7 gives the conditional grouping threshold parameter proxi for each example₀. For example, for the djsc500.5 example, a threshold of 0.9 is set, indicating that the condition is satisfied when the coloring layout similarity of the two parents reaches 90%, and color grouping is performed. This is because adding such a strategy may destroy the original solution structure, and the update of the coloring probability matrix with a certain randomness may easily break the structure that will converge to the final solution. And the grouping strategy is necessary to be dynamically adjusted according to the proxi value, and the correctness of dynamic grouping according to conditions can be proved by referring to the comparative experiment result given in the table 5.

According to experienceRendering layout similarity threshold proxi₀The value range of (2) is preferably 0.8-0.95, if the value of the coloring layout similarity threshold is too low, the original gradually-converged solution structure is easily damaged, so that the solution efficiency and quality are reduced, and if the value of the coloring layout similarity threshold is too high, the condition grouping strategy is difficult to trigger, which is not beneficial to keeping the population diversity.

4.4 lifting for le450_15c and le450_15d

It can be seen that the two graphs, le450_15c and le450_15d, are significantly improved in success rate compared to the head algorithm, and an analysis is made here. For such an extremely sparse graph (450 edges around the vertex 16000), if the HEAD algorithm is simply used for solving, the two individuals can rapidly converge, that is, the similarity of the coloring layout rapidly reaches the threshold value, so that the GPX intersection strategy is disabled and terminated. After the probability learning strategy is introduced, due to the fact that the coloring probability matrix is updated with certain randomness, the structure which is originally to be converged to a final solution is appropriately broken, the coloring layout similarity is reduced, meanwhile, the solution is continuously carried out on the original solution structure, and therefore the success rate is greatly improved on the example.

In summary, the mixed heuristic graph coloring method provided by the invention mainly introduces a probability learning strategy into a framework of an evolutionary algorithm to solve the graph coloring problem, and because only two coloring grouping schemes are provided, the diversity is rapidly reduced while the population is rapidly converged, and the individual is adjusted according to the coloring probability matrix. Under the condition of not influencing the population convergence speed, correct diversity can be introduced at proper time, and meanwhile, the settlement accuracy of the graph coloring grouping scheme is ensured. Through data example experiments, the improved scheme of the mixed heuristic graph coloring method has better effect on most data examples, so that the calculation precision and the calculation efficiency of the graph coloring problem are better balanced and considered.

Finally, it is noted that the above-mentioned embodiments illustrate rather than limit the invention, and that, while the invention has been described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims

1. A method for coloring a hybrid heuristic graph is characterized by comprising the following steps:

2. The method for coloring a hybrid heuristic graph according to claim 1, wherein in the step S2, the predetermined conditional grouping policy is specifically:

proxi＝q/n；

3. The hybrid heuristic graph coloring method of claim 2, in which the coloring layout similarity threshold proxi₀The value range of (A) is 0.8-0.95.

4. The hybrid heuristic graph coloring method of claim 1, wherein in the greedy cross-iteration of step S3, two coloring grouping schemes are labeled as a first parent and a second parent, respectively; in step S3, the greedy cross iteration method specifically includes:

5. The method of claim 4, wherein in step S301 and step S303, if there are more than one group with the highest number of vertices in the first parent, one group with the highest number of vertices in the first parent is randomly selected from the group.

6. The method for coloring a hybrid heuristic graph according to claim 4, wherein, in step S302, if there are a plurality of groupings with the highest number of vertices in the second parent, one of the groupings is randomly selected as the one with the highest number of vertices in the second parent.

7. The hybrid heuristic graph coloring method of claim 4, wherein the iteration termination condition is that a number of greedy cross iterations has reached a preset upper iteration number limit, or that the resulting new first parent is already a compliant coloring grouping scheme.

8. The method of claim 4, wherein the optimal solution of the greedy cross-iterated two coloring grouping schemes in step S5 is the first parent of the two coloring grouping schemes.

9. The method for coloring a hybrid heuristic graph according to claim 4, wherein in the step S4, the method for updating the color probability matrix is specifically as follows:

for a vertex in a group in the coloring grouping scheme: