CN112102438B - Graph dyeing problem searching method based on elite solution driven multi-level tabu search - Google Patents
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Abstract
The invention provides a graph dyeing problem searching method based on elite solution driving multi-level tabu search, which comprises the following steps: combining the two local search schemes to generate an initial elite solution with excellent performance; utilizing elite solution condensation current diagram with excellent performance, thereby sequentially obtaining smaller patterns of the next level; according to the weight, performing weight tabu dyeing on the intermediate graph of the condensation stage and the decondensation stage; gradually decondensing the condensation diagram to finally obtain an original diagram; and (5) jumping out of the local optimal area by adopting a disturbance method. According to the invention, a dynamic multi-level optimization scheme based on elite solution is adopted instead of a multi-level scheme based on a static graph network structure, and a weight tabu search method is designed at each level on the basis of the dynamic optimization scheme, the network structure and historical information, so that a solution with excellent graph dyeing problem performance is obtained. The invention has important research and practical values for the graph dyeing problem and the corresponding problem.
Description
Technical Field
The invention belongs to the technical field of artificial intelligence combination optimization, relates to a method for solving the graph dyeing problem, and in particular relates to a searching method for solving the graph dyeing problem based on multi-level tabu search driven by elite solution.
Background
The graph dyeing problem is a graph theory and an industrially very important research subject, so that the graph dyeing problem is developed and deeply researched, and has extremely high theoretical value and practical value. In particular, on the one hand, the graph staining problem is the most widely studied NP-hard problem, is computationally challenging, has extremely high research value, and is always a hot spot problem in the academia. On the other hand, the pattern dyeing problem is also a very active research topic in industry and has wide practical application value. The application studies of the current pattern dyeing in practical work include: nucleic acid sequence design in biochemistry, air traffic management, channel allocation in wireless networks, circuit design, scheduling problems, group detection in social networks, etc.
Faced with the computational challenges of graph-dyeing optimization problems, as well as the inherent difficulties of the problems themselves, many researchers have focused on developing approximation or heuristic algorithms. However, existing heuristic algorithms suffer from two drawbacks in dealing with graph staining problems:
(1) Only a single-layer processing mode is adopted. Typical heuristic algorithms in the prior art include construction algorithms, local search algorithms, population-based algorithms, etc., however, the above algorithms are all single-level processing methods. Very little research has been done on the graph staining problem for multi-level (also known as multi-scale) optimization methods.
(2) Dynamic weighting techniques are not designed to preserve historical information. As a variant of the local search algorithm, a reasonably designed weighting technique can dynamically reflect the historical search situation of the solution, and a reasonably combined weighting technique and multi-level optimization method can achieve better balance between development and exploration of each level. Weighting techniques have been successfully applied to many classical NP-complete problems such as aggregate coverage, minimum vertex coverage, and satisfiability problems.
It is known that only one early undisclosed algorithm uses a multi-level approach to solving the graph staining problem, but the results are largely undesirable because it uses only a general multi-level approach and does not design a special multi-level strategy for the graph staining problem. The invention re-researches the multi-level scheme of graph dyeing by adopting a higher-level strategy and weighting technology in a multi-level key component, and aims to provide a quick heuristic local search method with excellent performance.
Disclosure of Invention
Based on the above, the invention provides a solution with excellent performance of graph dyeing problem, which further improves dyeing performance by utilizing the structure of the information reduction graph contained in elite solution and a weight tabu search scheme. The invention can better solve the problem of graph dyeing, and has higher applicability to network structures of different graphs. According to the invention, a dynamic multi-level optimization scheme based on elite solution is adopted instead of a multi-level scheme based on a static graph network structure, and a weight tabu search method is designed at each level on the basis of the dynamic optimization scheme, the network structure and historical information, so that a solution with excellent graph dyeing problem performance is obtained. The invention has important research and practical values for the graph dyeing problem and the corresponding problem.
In order to achieve the above purpose, the present invention provides the following technical solutions:
the graph dyeing problem searching method based on elite solution driven multi-level tabu search comprises the following steps:
(1) Combining two local search schemes generates an initial elite solution with excellent performance: given diagram G 0 ＝(V 0 ,E 0 ) And the dyeing number k, firstly, generating an initial graph dyeing scheme with smaller conflict by using a greedy method, and further reducing the conflict number by using an iterative tabu search mode to improve the quality of an initial solution so as to generate an initial elite solution;
(2) The current graph is condensed by elite solution with excellent performance, so that smaller graphs of the next level are sequentially obtained: the condensation stage inputs the graph G by merging vertices 0 Converting into a series of monotonously decreasing coarsening patterns, assuming m as the current level, condensing pattern G according to the current solution m Conversion to smaller graphic G m+1 ；
(3) Weight tabu staining is performed on the intermediate graph of the condensation stage and the decondensation stage according to the weights: in the condensation stage, a dyeing scheme is firstly generated by inheriting the solution of the m layer; then applying a weight tabu staining algorithm to further improve the staining protocol; finally, the condensation phase uses the improved solution for creating the next condensation graph;
(4) Gradually decondensing the condensation diagram to finally obtain an original diagram: when the condensation stage reaches the last layer L, the decondensation stage is triggered, i.e. the current condensation pattern G m Expanded into more expanded graph G m-1 The method comprises the steps of carrying out a first treatment on the surface of the Then, the condensation diagram G m Dyeing scheme S of (2) m G projected to the last level m-1 Dyeing scheme S of (2) m-1 Then, the dyeing scheme S is subjected to a weight tabu dyeing algorithm m-1 Optimizing and improving; the above-mentioned stages are circulated until the original graph G is restored 0 The method comprises the steps of carrying out a first treatment on the surface of the Wherein the decondensation stage resumes L intermediate graphs in reverse order to the condensation stage until G 0 Applying a weight tabu staining algorithm to each layer of the decondensed graph to improve the quality of the solution;
(5) And (3) jumping out of a local optimal area by adopting a disturbance method: when the decondensation stage is terminated, if for G 0 Improved, then move to the next round of multi-level processing; otherwise, if the solution cannot be improved by multiple successive levels, the search is considered to be trapped in a locally optimal trap, triggering a perturbation phase to alter some colors with specific identified vertices; the perturbed solution is used as an initial solution for the next layer solution.
Further, the step (1) specifically includes the following steps:
for a given graph G 0 ＝(V 0 ,E 0 )，V 0 Is the vertex, E 0 For the edge, the number of colors is k, firstly, a DANGER greedy algorithm is adopted to establish a k-dyeing scheme S with the least conflict as possible 0 : selecting a vertex allocation color from a group of non-allocated dyed vertices at a time, and firstly scoring the vertices of the non-allocated colors by using a dynamic scoring function; based on these scoring scores, probabilistically selecting the unassigned vertex with the highest score; calculating the score of the corresponding color of the vertex according to the possibility that the adjacent vertex needs the dyeing; selecting a color for the vertex probabilistically based on the color score; repeating the above process until all vertices are assigned colors; then, adopting tabu search pairsOptimizing dyeing scheme, reducing conflict number, improving initial solution quality, generating a new elite solution S 0 。
Further, the step (2) specifically includes the following sub-steps:
(a) Initializing the chart G to be condensed m+1 : at the beginning of condensation, the condensation diagram G of the m+1th level m+1 Set to G m Solution S m+1 Set as S m Side w m+1 The weight of (2) is set to w m ；
(b) Increasing the weight of edges between conflicting vertices: dynamically increasing the weight of edges between conflict vertexes by adopting a heuristic strategy;
(c) Condensing non-conflicting vertices: when the vertex v m+1 And vertex u m+1 Is a non-conflicting vertex assigned to the same vertex set, then probabilistically condensing vertex v m+1 And vertex u m+1 And update and vertex v m+1 And vertex u m+1 Weighting of adjacent edges; this process is repeated until all segments are traversed.
Further, the step (c) specifically includes the following sub-steps:
i) The probability of each fragment being condensed was evaluated: at the m+1 level, for each pair of vertices u m+1 And vertex v m+1 Defining a condensation scoring function score [ u ] m+1 ][v m+1 ]The method comprises the steps of carrying out a first treatment on the surface of the Let Record [ u ] m+1 ][v m+1 ]Vertex u in the solution of elite in the past m+1 And v m+1 SumRecord is the total number of elite solutions, then vertex u m+1 And vertex v m+1 The condensation scoring function between is defined as:
score[u m+1 ][v m+1 ]＝(Record[u m+1 ][v m+1 ]+2)/(SumRecord+3)
wherein u is 1 to or less m ,v m ≤N；
ii) probabilistic selective condensation of vertices: probabilistic determination of vertices v based on a condensation scoring function for each pair of vertices m+1 And u m+1 Whether or not condensed, vertex v m+1 And u m+1 Condensed cocoaThe energy is represented by rand [0,1 ]]Is given;
iii) Updating the weight of the condensed vertex: condensation diagram G at m+1 level m+1 Edge (v) m+1 ,u m+1 ) Weights equal to the mth level graph G m The sum of the weights of the edges of the corresponding vertices of (a).
Further, the step (3) specifically includes the following steps:
first defining search space and evaluation function: for a given graph G ' = (V ', E ') with k available colors, the search space contains all possible k candidate colors; candidate staining protocol in Ω 'was performed with S' = { V 1 ′,V 2 ′,...,V k ' represents, wherein V i ' is a set of vertices that receive the same color i;
wherein i is not equal to j,1 is not less than i, j is not less than k;
the evaluation function f ' (S ') is used to calculate the sum of the weights of the conflict sides in the dyeing allocation scheme S ', the formula of which is as follows
Wherein, |C w (V i ' is color category V i The weight of the' middle conflict edge; thus, the dyeing scheme S ' having f ' (S ')=0 corresponds to k dyeing by the condensation method;
given a conflicting k-dyeing assignment scheme S' = { V 1 ′,V 2 ′,...,V k ' the basic idea of a movement neighborhood N (S ') is to move a conflicting vertex V ' from its original set of vertices V i 'move to another subset V' j The method comprises the steps of carrying out a first treatment on the surface of the A matrix B of size n x k is used, wherein the elements B [ v ]'][i]Record and color class V i V 'the number of adjacent vertices in' where 1.ltoreq.i.ltoreq.k; the moving gain of the optimization objective function variation is expressed as follows:
Δf′＝B[v′][j]-B[v′][i]
each time a single step operation involving vertex v' is performed, only a subset of the values affected by this movement need be updated; for each vertex u 'adjacent to vertex v', there is B [ u '] [ i ] [ B [ u' ] [ i ] -w (v ', u'), and B [ u '] [ j ] [ B [ u' ] [ j ] +w (v ', u');
weight tabu search the best neighbor S is selected according to the mobile gain and tabu list calculated from the mobile gain formula Nbest E N (S'), where S Nbest Is the minimum value that is not in the tabu list or is better than the best solution found so far; when there are multiple vertices with the same movement gain, one vertex movement is randomly selected.
Further, the detailed decondensation step from the m-th level to the m-1 th level in the step (4) comprises the following sub-steps:
(a) Will G m-1 Initialized to G m Dyeing scheme S m-1 Initialized to S m Weight of edge w m-1 Initialized to w m ；
(b) Unfolding the condensed vertices and updating the corresponding solution and edge weights: expanding the condensed vertices at the mth level to form a condensed graph G at the mth-1 level m-1 The method comprises the steps of carrying out a first treatment on the surface of the Accordingly, staining protocol S of the mth hierarchy m Dyeing scheme S extending to the m-1 th level m-1 The method comprises the steps of carrying out a first treatment on the surface of the In the decondensation stage, each pair of condensed vertices is decondensed and assigned the same color; finally, updating the edge weight of the unfolded vertex;
(c) Increasing the weight of edges between conflicting vertices: after decondensation of the two vertices, if vertex v m-1 And u m-1 Is a conflicting vertex, a larger value is added to its base edge weight.
Further, the disturbing step (5) includes the following operations:
moving a certain number of vertices using a single point move operator, the move not being able to move back to the original color in the next tts moves; the tabu length of the perturbation phase is tt=f+rand (1,1000), where f is the number of vertex collisions; the dyeing scheme obtained in the disturbance process is used as an initial dyeing scheme of the next round of multi-level stages.
Compared with the prior art, the invention has the following advantages and beneficial effects:
according to the invention, a dynamic multi-level optimization scheme based on elite solution is adopted instead of a multi-level scheme based on a static graph network structure, and a weight tabu search method is designed at each level on the basis of the dynamic optimization scheme, the network structure and historical information, so that a solution with excellent graph dyeing problem performance is obtained. The invention has important research and practical values for the graph dyeing problem and the corresponding problem.
Drawings
FIG. 1 is a flow chart for solving for k-staining based on an elite solution driven multi-level weight tabu search method.
FIG. 2 is an example of an elite solution-based condensation process, where (a) represents a condensation diagram G 1 Is initialized to the original graph G 0 The method comprises the steps of carrying out a first treatment on the surface of the (b) increasing the weight of the conflict-side in the condensation graph; (c) One possible way to upgrade the weight of non-conflict edges during condensation is represented.
FIG. 3 is an example of a weight tabu search process with tabu vertices;
FIG. 4 is an example of a decondensation process in which (a) represents the initialization process of the decondensation stage, G 0 ＝G 1 ，S 0 ＝S 1 ，w 0 ＝w 1 (b) represents the diagram G 0 Restoring the condensed vertices and upgrading the corresponding edges, (c) increasing the weight of the conflict edges in the decondensation graph.
Detailed Description
The technical scheme provided by the present invention will be described in detail with reference to the following specific examples, and it should be understood that the following specific examples are only for illustrating the present invention and are not intended to limit the scope of the present invention.
The invention provides a graph dyeing problem searching method based on elite solution driven multi-level tabu search, which specifically adopts a scheme shown in a figure 1 and comprises the following steps:
(1) An initial solution with excellent performance is generated. For a given graph G 0 ＝(V 0 ,E 0 )，V 0 Is the vertex, E 0 For the edge, the number of colors k, a dyeing scheme (solution) with excellent performance is firstly generated, and the process is the initialization of the dyeing process. Specifically, a DANGER greedy algorithm is first used to establish a k-dyeing scheme S with as little collisions as possible 0 . One vertex at a time is selected from a set of unassigned vertices to assign color. Specifically, vertices of unassigned colors are first scored using a dynamic scoring function. Based on these scoring scores, the unassigned vertex with the highest score is probabilistically selected. Then, the score of the corresponding color for the vertex is calculated based on the likelihood that the neighboring vertex requires this coloring. Finally, a color is probabilistically selected for the vertex based on the color score. The above process is repeated until all vertices are assigned colors. In order to further optimize the solution constructed by the DANGER strategy, the invention adopts tabu search to optimize the dyeing scheme, reduces the conflict number, improves the quality of the initial solution and generates a new elite solution S 0 . In the initialization process, improving the quality of the initial elite solution plays a vital role in the multi-level condensation process based on the elite solution.
(2) And designing a multi-level condensation mode according to elite solution. Since the algorithm used to dye each condensed graph is a heuristic, some vertices may be incorrectly assigned the same color and thus incorrectly merged into one vertex. To solve this problem, we devised a condensation strategy for heuristic multi-level graph staining that probabilistically merges collision-free vertices with the same color during the condensation phase. Let m be the current level, let G m ＝(V m ,E m ) For the condensation diagram of the m-th hierarchy, the condensation stage will be diagram G m Condensation to form the graph G m+1 . In the condensation stage, the reaction mixture is prepared by m The operation of middle edges and vertices will seriously affect the next level condensation graph G m+1 Is characterized by comprising the following components in parts by mass: to jump out of local optima of the mth level, the condensation phase increases the weight of conflicting edges; according to the dynamically learned information, fragments composed of the m-th level non-conflict vertexes are probabilistically condensed to the m+1th level vertexes. Conflicting vertices or not belonging to any constrictionNon-conflicting vertices of the synthetic segment, i.e., non-condensed vertices, will simply be replicated to G m+1 . The method specifically comprises the following substeps:
(a) Initializing the chart G to be condensed m+1 . At the beginning of condensation, the condensation diagram G of the m+1th level m+1 Set to G m Solution S m+1 Set as S m Side w m+1 The weight of (2) is set to w m . FIG. 2 provides an original graph G with 5 vertices 0 Condensation into condensation diagram G of level 1 1 Is an example of (a). As shown in FIG. 2 (a), the 1 st level of condensation diagram G 1 Is initialized to G 0 。
(b) The weights of the edges between conflicting vertices are increased. During the weight tabu search, the weight of the edge affects the generation of the staining solution on the m+1 level. In order to leave the locally optimal region of the mth level, the weights of the edges are dynamically adjusted during the condensation phase. In order to avoid the same collision of vertexes and facilitate searching to get rid of local optimization of an mth level, the invention adopts a heuristic strategy to dynamically increase the weight of edges between the collision vertexes. As shown in FIG. 2 (b), the condensation pattern G is increased 1 Conflicting vertex 2 in (1) 1 And 3 1 Weight of condensed edge between w (2 1 ,3 1 )，Where w (x, y) is the original graph G 0 The edge weights of vertices x and y in the original graph, N is the number of vertices in the original graph.
(c) Condensing non-conflicting vertices. If the vertex v m+1 And vertex u m+1 Is a non-conflicting vertex assigned to the same vertex set, then probabilistically condenses vertex v m+1 And vertex u m+1 And update and vertex v m+1 And vertex u m+1 Weighting of adjacent edges. This process is repeated until all segments are traversed. This step can be broken down into three parts:
evaluate the likelihood that each fragment is condensed. At the m+1 level, for each pair of vertices u m+1 And vertex v m+1 Defining a condensation scoring function score [ u ] m+1 ][v m+1 ]. Let Record [ u ] m+1 ][v m+1 ]Vertex u in the solution of elite in the past m+1 And v m+1 SumRecord is the total number of elite solutions, then vertex u m+1 And vertex v m+1 The condensation scoring function between is defined as:
score[u m+1 ][v m+1 ]＝(Record[u m+1 ][v m+1 ]+2)/(SumRecord+3)
wherein u is 1 to or less m ,v m ≤N。
Probabilistic selection of condensed vertices. Probabilistic determination of vertices v based on a condensation scoring function for each pair of vertices m+1 And u m+1 Whether or not to be condensed. Vertex v m+1 And u m+1 The possibility of condensation is defined by rand [0,1]Given.
Update the weights of the condensed vertices. Condensation diagram G at m+1 level m+1 Edge (v) m+1 ,u m+1 ) Weights equal to the mth level graph G m The sum of the weights of the edges of the corresponding vertices of (a).
Fig. 2 (c) shows possible condensation of non-conflicting vertices during the condensation process: suppose vertex 1 0 And 4 0 Score of (2) is greater than random probability, G 1 Vertex 1 of (2) 1 By folding G 0 Vertex 1 of (2) 0 And 4 0 And (3) forming a vertex. Thereafter, these initial graphs G 0 Middle and vertex 1 0 And 4 0 The edges between the associated vertices are condensed into a new edge, the weights of which are set to their endpoints with 1 0 And 4 0 The sum of the weights of the associated edges, i.e. w (1 1 ,5 1 )＝w(1 0 ,5 0 )+w(4 0 ,5 0 )＝2，w(1 1 ,3 1 )＝w(1 0 ,3 0 )+w(4 0 ,3 0 ) =2, wherein w (u m ,v m ) Representation of diagram G m Middle vertex u m And v m Edge weights in between.
We note that for non-conflicting vertices, this condensation rule is effective to repair the misclassification of non-conflicting vertices of the same color.
The invention sets the coarsening operation to repeat L times (parameters, default set to 5), thereby generating L condensation graphs.
Will G m Condensation to G m+1 After that, the invention applies weight tabu search to improve the condensation graph G m+1 The weight tabu search helps the search reach a promising search range quickly.
(3) The weight tabu search promotes the staining scheme of each layer of condensation map. The purpose of the condensation stage is to pass through the m-th level of graph G m Obtaining a graph G of an m+1 th level condensation m+1 The purpose of the decondensation stage is to pass G m Obtaining an expanded m-1 level expansion graph G m-1 . The weight tabu dyeing algorithm is designed to improve the quality of solutions after the condensation stage and the decondensation stage. For clarity, the present invention uses G ' = (V ', E ') to represent the newly generated map (i.e., G of the condensation stage) m+1 G in the decondensation stage m-1 ) S' as the corresponding solution.
The weight iterative tabu search process will further improve the quality of the solution S', generating an elite solution for the current level, ready for the next condensation (or decondensation) process. The basic operation of the weight iterative tabu search scheme is as follows: a search space and an evaluation function are first defined. For a given graph G ' = (V ', E ') with k available colors, the search space contains all possible k colors (candidate colors). Candidate staining protocols in Ω 'can be used with S' = { V 1 ′,V 2 ′,...,V k ' represents, wherein V i ' is a set of vertices that receive the same color i.
Wherein i.noteq.j, 1.ltoreq.i, j.ltoreq.k.
The evaluation function f ' (S ') is used to calculate the sum of the weights of the conflict sides in the dyeing allocation scheme S ', the formula of which is as follows
Wherein, |C w (V i ' is color category V i The sum of the weights of the 'middle conflict' edges. Thus, the dyeing scheme S ' having f ' (S ')=0 corresponds to k dyeing by the condensation method.
Given a conflicting k-dyeing assignment scheme S' = { V 1 ′,V 2 ′,...,V k ' the basic idea of a movement neighborhood N (S ') is to move a conflicting vertex V ' from its original set of vertices V i 'move to another subset V' j . We use a matrix B of size n x k, where the elements B v'][i]Record and color class V i V 'the number of adjacent vertices in' 1.ltoreq.i.ltoreq.k. The movement gain is expressed as follows:
Δf′＝B[v′][j]-B[v′][i]
according to the fast delta computation technique, only a subset of the values affected by this movement need be updated each time a single step operation involving vertex v' is performed. For each vertex u ' adjacent to vertex v ', there is B [ u ] '][i]←B[u′][i]-w v′,u′ And Bu'][j]←B[u′][j]+w v′,u′ 。
The best neighbor S is selected by the movement gain and the tabu list calculated by the weight tabu search Nbest E N (S'), where S Nbest Is the minimum value that is not in the tabu list (i.e. S Nbest Is the best neighborhood solution that is not prohibited) or is better than the best solution found so far. If there are multiple vertices with the same movement gain, one vertex movement is randomly selected.
To illustrate the improved process of weight tabu search, we consider the condensed graph G of the first level of the graph in FIG. 2 (c) 1 And 3-staining as shown in FIG. 2 (c). Fig. 3 shows one possible scenario of the weight dyeing process: suppose when vertex 3 1 In the tabu list, the staining algorithm cannot find G 1 Is a legal 3-dyeing of (2), vertex 2 1 The staining was altered to obtain the optimization objective f' =1.
(4) A multi-stage decondensation stage. The decondensation stage is the reverse of the condensation stage, i.e. from the condensationDrawing G m Middle stage by stage restoration graph G m-1 Until the initial graph G is reached 0 . Specifically, the detailed steps from the mth level to the m-1 th level are as follows:
(a) Will G m-1 Initialized to G m Dyeing scheme S m-1 Initialized to S m Weight of edge w m-1 Initialized to w m 。
(b) The condensed vertices are expanded and the corresponding solution and edge weights are updated. Expanding the condensed vertices at the mth level to form a condensed graph G at the mth-1 level m-1 . Accordingly, staining protocol S of the mth hierarchy m Dyeing scheme S extending to the m-1 th level m-1 . Since in the condensation stage, each pair of condensed vertices is not adjacent and has the same coloration, in the decondensation stage, the present invention will decondense these vertices and assign them the same coloration. And finally, updating the edge weight of the unfolded vertex.
(c) The weights of the edges between conflicting vertices are increased. After decondensation of the two vertices, if vertex v m-1 And u m-1 Is a conflicting vertex, a larger value is added to its base edge weight.
After the decondensation stage, a weight tabu search strategy is adopted, and the solution is further improved through the updated weight information in the decondensation process.
An example of a decondensation stage is shown in figure 4. FIG. 4 (a) shows a first level of condensation diagram G 1 Vertex 1 of (2) 1 Recovered process, the vertex is formed by G in the condensation stage 0 Vertex 1 of (2) 0 And 4 0 And (3) folding. Thus, vertex 1 0 And 4 0 All of the dyeings of (1) 1 And vertex 1 0 And 4 0 The edge weights in between are restored, i.e. w (1 0 ,3 0 )＝w(1 0 ,5 0 )＝1，w(4 0 ,3 0 )＝w(4 0 ,5 0 ) =1. Finally, add conflicting vertex 1 0 And 2 0 The weight of the condensed edge between them,
(5) The perturbation phase helps the search process jump out of the locally optimal region. Heuristic tabu search processes focus searches only around conflicting vertices and thus may fall into a locally optimal solution. Therefore, we apply simple perturbations when the weight tabu falls into a local optimum. The perturbation uses a single point move operator to move a certain number of vertices (set to 0.1 x n in the present invention). To avoid excessive degradation of the perturbation solution, the perturbation considers the fitness value and cannot be shifted back to the original color in the next tts of shifts, where tt represents the tabu length of the perturbation phase and is defined in this invention as tt=f+rand (1,1000) (f is the number of vertex collisions). Then, the dyeing scheme obtained in the disturbance process is used as an initial dyeing scheme of the next round of multi-level stages.
The technical means disclosed by the scheme of the invention is not limited to the technical means disclosed by the embodiment, and also comprises the technical scheme formed by any combination of the technical features. It should be noted that modifications and adaptations to the invention may occur to one skilled in the art without departing from the principles of the present invention and are intended to be within the scope of the present invention.
Claims (4)
1. The graph dyeing problem searching method based on elite solution driven multi-level tabu search is characterized by comprising the following steps of:
(1) Combining two local search schemes generates an initial elite solution with excellent performance: given diagram G 0 ＝(V 0 ,E 0 ) And the dyeing number k, firstly, generating an initial graph dyeing scheme with smaller conflict by using a greedy method, and further reducing the conflict number by using an iterative tabu search mode to improve the quality of an initial solution so as to generate an initial elite solution;
(2) The current graph is condensed by elite solution with excellent performance, so that smaller graphs of the next level are sequentially obtained: the condensation stage inputs the graph G by merging vertices 0 Converting into a series of monotonously decreasing coarsening patterns, assuming m as the current level, and condensing the patterns according to the current solutionShape G m Conversion to smaller graphic G m+1 The method specifically comprises the following substeps:
(a) Initializing the chart G to be condensed m+1 : at the beginning of condensation, the condensation diagram G of the m+1th level m+1 Set to G m Solution S m+1 Set as S m Side w m+1 The weight of (2) is set to w m ；
(b) Increasing the weight of edges between conflicting vertices: dynamically increasing the weight of edges between conflict vertexes by adopting a heuristic strategy;
(c) Condensing non-conflicting vertices: when the vertex v m+1 And vertex u m+1 Is a non-conflicting vertex assigned to the same vertex set, then probabilistically condensing vertex v m+1 And vertex u m+1 And update and vertex v m+1 And vertex u m+1 Weighting of adjacent edges; repeating this process until all segments are traversed; the step (c) specifically comprises the following sub-steps:
i) The probability of each fragment being condensed was evaluated: at the m+1 level, for each pair of vertices u m+1 And vertex v m+1 Defining a condensation scoring function score [ u ] m+1 ][v m+1 ]The method comprises the steps of carrying out a first treatment on the surface of the Let Record [ u ] m+1 ][v m+1 ]Vertex u in the solution of elite in the past m+1 And v m+1 SumRecord is the total number of elite solutions, then vertex u m+1 And vertex v m+1 The condensation scoring function between is defined as:
score[u m+1 ][v m+1 ]＝(Record[u m+1 ][v m+1 ]+2)/(SumRecord+3)
wherein u is 1 to or less m ,v m ≤N；
ii) probabilistic selective condensation of vertices: probabilistic determination of vertices v based on a condensation scoring function for each pair of vertices m+1 And u m+1 Whether or not condensed, vertex v m+1 And u m+1 The possibility of condensation is defined by rand [0,1]Is given;
iii) Updating the weight of the condensed vertex: condensation diagram G at m+1 level m+1 Edge (v) m+1 ,u m+1 ) Is equal to the weight of the (1)m-level graph G m The sum of the weights of the edges of the corresponding vertices of (a);
(3) Weight tabu staining is performed on the intermediate graph of the condensation stage and the decondensation stage according to the weights: in the condensation stage, a dyeing scheme is firstly generated by inheriting the solution of the m layer; then applying a weight tabu staining algorithm to further improve the staining protocol; finally, the condensation phase uses the improved solution for creating the next condensation graph; the method specifically comprises the following steps:
first defining search space and evaluation function: for a given graph G ' = (V ', E ') with k available colors, the search space contains all possible k candidate colors; candidate staining protocol in Ω 'was performed with S' = { V 1 ′,V′ 2 ,...,V′ k Represented by V where i ' is a set of vertices that receive the same color i;
wherein i is not equal to j,1 is not less than i, j is not less than k;
the evaluation function f ' (S ') is used to calculate the sum of the weights of the conflict sides in the dyeing allocation scheme S ', the formula of which is as follows
Wherein, |C w (V i ' is color category V i The weight of the' middle conflict edge; thus, the dyeing scheme S ' having f ' (S ')=0 corresponds to k dyeing by the condensation method;
given a conflicting k-dyeing assignment scheme S' = { V 1 ′,V′ 2 ,…,V′ k The basic idea of a moving neighborhood N (S ') is to move a conflicting vertex V' from its original vertex set V i 'move to another subset V' j The method comprises the steps of carrying out a first treatment on the surface of the A matrix B of size n x k is used, wherein the elements B [ v ]'][i]Record and color class V i V 'number of adjacent vertices in' whereinI is more than or equal to 1 and less than or equal to k; the moving gain of the optimization objective function variation is expressed as follows:
Δf′＝B[v′][j]-B[v′][i]
each time a single step operation involving vertex v' is performed, only a subset of the values affected by this movement need be updated; for each vertex u 'adjacent to vertex v', there is B [ u '] [ i ] [ B [ u' ] [ i ] -w (v ', u'), and B [ u '] [ j ] [ B [ u' ] [ j ] +w (v ', u');
weight tabu search the best neighbor S is selected according to the mobile gain and tabu list calculated from the mobile gain formula Nbest E N (S'), where S Nbest Is the minimum value that is not in the tabu list or is better than the best solution found so far; randomly selecting one vertex to move when the vertices have a plurality of same movement gains;
(4) Gradually decondensing the condensation diagram to finally obtain an original diagram: when the condensation stage reaches the last layer L, the decondensation stage is triggered, i.e. the current condensation pattern G m Expanded into more expanded graph G m-1 The method comprises the steps of carrying out a first treatment on the surface of the Then, the condensation diagram G m Dyeing scheme S of (2) m G projected to the last level m-1 Dyeing scheme S of (2) m-1 Then, the dyeing scheme S is subjected to a weight tabu dyeing algorithm m-1 Optimizing and improving; the above-mentioned stages are circulated until the original graph G is restored 0 The method comprises the steps of carrying out a first treatment on the surface of the Wherein the decondensation stage resumes L intermediate graphs in reverse order to the condensation stage until G 0 Applying a weight tabu staining algorithm to each layer of the decondensed graph to improve the quality of the solution;
(5) And (3) jumping out of a local optimal area by adopting a disturbance method: when the decondensation stage is terminated, if for G 0 Improved, then move to the next round of multi-level processing; otherwise, if the solution cannot be improved by multiple successive levels, the search is considered to be trapped in a locally optimal trap, triggering a perturbation phase to alter some colors with specific identified vertices; the perturbed solution is used as an initial solution for the next layer solution.
2. The graph dyeing problem searching method based on elite solution driven multi-level tabu search according to claim 1, wherein the step (1) specifically comprises the following steps:
for a given graph G 0 ＝(V 0 ,E 0 )，V 0 Is the vertex, E 0 For the edge, the number of colors is k, firstly, a DANGER greedy algorithm is adopted to establish a k-dyeing scheme S with the least conflict as possible 0 ；
Selecting a vertex allocation color from a group of non-allocated dyed vertices at a time, and firstly scoring the vertices of the non-allocated colors by using a dynamic scoring function; based on these scoring scores, probabilistically selecting the unassigned vertex with the highest score; calculating the score of the corresponding color of the vertex according to the possibility that the adjacent vertex needs the dyeing; selecting a color for the vertex probabilistically based on the color score; repeating the above process until all vertices are assigned colors;
adopting tabu search to optimize the dyeing scheme, reducing conflict number, improving the quality of initial solution and generating a new elite solution S 0 。
3. The graph dyeing problem search method based on elite solution driven multi-level tabu search according to claim 1, wherein the detailed decondensation step from the mth level to the m-1 th level in the step (4) comprises the following sub-steps:
(a) Will G m-1 Initialized to G m Dyeing scheme S m-1 Initialized to S m Weight of edge w m-1 Initialized to w m ；
(b) Unfolding the condensed vertices and updating the corresponding solution and edge weights: expanding the condensed vertices at the mth level to form a condensed graph G at the mth-1 level m-1 The method comprises the steps of carrying out a first treatment on the surface of the Accordingly, staining protocol S of the mth hierarchy m Dyeing scheme S extending to the m-1 th level m-1 The method comprises the steps of carrying out a first treatment on the surface of the In the decondensation stage, each pair of condensed vertices is decondensed and assigned the same color; finally, updating the edge weight of the unfolded vertex;
(c) Adding conflict roofWeighting of edges between points: after decondensation of the two vertices, if vertex v m-1 And u m-1 Is a conflicting vertex, a larger value is added to its base edge weight.
4. The graph dyeing problem search method based on elite solution driven multi-level tabu search according to claim 1, wherein the perturbation phase in step (5) comprises the following operations:
moving a certain number of vertices using a single point move operator, the move not being able to move back to the original color in the next tts moves; the tabu length of the perturbation phase is tt=f+rand (1,1000), where f is the number of vertex collisions; the dyeing scheme obtained in the disturbance process is used as an initial dyeing scheme of the next round of multi-level stages.
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Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2001022726A (en) * | 1999-07-08 | 2001-01-26 | Fujitsu Ltd | Taboo searching device |
| CN106610640A (en) * | 2015-11-23 | 2017-05-03 | 四川用联信息技术有限公司 | Tabu list containing genetic and local search algorithm for multi-objective flexible job-shop scheduling |
| CN107730568A (en) * | 2017-10-31 | 2018-02-23 | 山东师范大学 | Color method and device based on weight study |
| CN110110753A (en) * | 2019-04-03 | 2019-08-09 | 河南大学 | Effective hybrid characteristic selecting method of pollination algorithm and ReliefF is spent based on elite |
-
2020
- 2020-08-26 CN CN202010872880.7A patent/CN112102438B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2001022726A (en) * | 1999-07-08 | 2001-01-26 | Fujitsu Ltd | Taboo searching device |
| CN106610640A (en) * | 2015-11-23 | 2017-05-03 | 四川用联信息技术有限公司 | Tabu list containing genetic and local search algorithm for multi-objective flexible job-shop scheduling |
| CN107730568A (en) * | 2017-10-31 | 2018-02-23 | 山东师范大学 | Color method and device based on weight study |
| CN110110753A (en) * | 2019-04-03 | 2019-08-09 | 河南大学 | Effective hybrid characteristic selecting method of pollination algorithm and ReliefF is spent based on elite |
Non-Patent Citations (2)
| Title |
|---|
| A solution-driven multilevel approach for graph coloring;Wen Sun等;Applied Soft Computing Journal;第104卷(第107期);1-12 * |
| 基于图染色问题的混合优化算法;李淑芝等;计算机应用研究;33(1);98-100 * |
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