Efficient distributed vertex cover method based on nash equilibrium selection

Abstract

The application relates to a high-efficiency distributed vertex covering method based on Nash equilibrium selection, which comprises the following steps: taking the sum of vertex weights and the punishment for uncovered edges as indexes, establishing a global optimization model of a minimum weighted vertex covering problem in a distributed network system; converting the global optimization model into a distributed collaborative decision model based on a network ordinal potential game; determining a neighbor set of each vertex in the distributed network system and initializing actions; obtaining neighbor action information through communication between each vertex and the neighbors, and iteratively updating the actions of each vertex until the individual actions of all the vertices no longer change, so that an optimized Nash equilibrium covering scheme is obtained. According to the scheme of the embodiment of the application, the approximate optimal scheme of the minimum weighted vertex covering problem which is independent of centralized nodes and global information is efficiently solved in a distributed environment, and efficient collaborative vertex covering is realized by only using neighborhood information and local interaction.

Description

TECHNICAL FIELD

[0001] The present application relates to the cross field of artificial intelligence, game theory and complex network, and particularly relates to an efficient distributed vertex cover method based on Nash equilibrium selection. BACKGROUND

[0002] As a classical problem in graph theory, the minimum weighted vertex cover considers an arbitrary simple undirected graph. For any subset of its point set, if it contains at least one vertex of all edges, the subset is called a vertex cover. The minimum weighted vertex cover problem aims to find the cover scheme with the minimum sum of weights from all vertex cover solutions. When the weights of nodes are equal, the problem degenerates into the minimum vertex cover problem. The minimum weighted vertex cover is one of the 21 NP-hard problems proposed by Karp, and has important theoretical significance in computational complexity science. In addition, the problem has important practical significance in engineering application fields such as road monitoring, computational biology, network security, very large scale integrated circuit design, and cooperative task allocation. Taking road monitoring as an example, it is assumed that in order to monitor the traffic situation of the road, some intersections are selected to install monitoring cameras to achieve the purpose of covering all branch roads. All cameras can monitor all adjacent branch roads, but considering different geographical environments and economic factors, different costs are required to set up monitoring cameras at different intersections. Therefore, by solving the optimal or suboptimal solution of the minimum weighted vertex cover of the traffic road network, the minimization or sub-minimization of the cost can be achieved.

[0003] As one of the well-known combinatorial optimization problems, the minimum weighted vertex cover has been proved to have NP-hard computational complexity. Unless P=NP, it cannot be accurately solved in polynomial time. Therefore, researchers have shifted their focus to developing efficient approximate solution algorithms. However, most existing methods are essentially centralized, requiring the use of a centralized control node and global information to supervise the entire optimization process, which has the disadvantages of poor robustness, vulnerability to attack, and exponential explosion. More importantly, in most distributed network systems, the existence of a centralized control node cannot be satisfied. On the contrary, each node can only obtain local neighborhood information. Therefore, the existing centralized solution method cannot be applied to large-scale, decentralized distributed network systems. Therefore, how to design distributed coordination rules to achieve approximate solution of the minimum weighted vertex cover under the conditions of limited local communication and autonomous negotiation has become an important research problem in the field of graph theory and distributed network systems.

[0004] In recent years, game theory methods have been increasingly applied in the distributed coordination and optimization of autonomous agent network systems. A distributed optimization method based on memory and best response is provided by regarding each vertex as an intelligent agent participating in the snowdrift game to solve the vertex covering problem in the unweighted version. However, this method is designed specifically for the unweighted version and cannot be extended to the generalized weighted version. Foreign scholars have studied the weighted vertex (set) covering problem in the potential game framework and proposed a cooperative algorithm with strong convergence. Unfortunately, this is achieved by integrating a carefully selected suggestion vector into the distributed behavior rule. This non-fully-distributed algorithm further emphasizes the fact that even in the potential game framework, it is difficult to obtain an excellent approximate solution without the help of a central control node and global information. Therefore, the real challenge lies in how to choose a fine Nash equilibrium solution with good global evaluation indicators from numerous Nash equilibrium solutions, rather than an arbitrary equilibrium solution that cannot guarantee the global system performance.

[0005] In summary, most of the existing solutions for the weighted minimum vertex covering problem belong to approximate optimization algorithms under the centralized communication architecture and cannot be applied to problem solving in a distributed environment. In addition, a few methods can use game theory as a tool to achieve Nash equilibrium solving in a distributed environment, but cannot guarantee further improvement and improvement of global performance. SUMMARY

[0006] To overcome the defects in the prior art described above, the purpose of the present application is to provide an efficient distributed vertex covering method based on Nash equilibrium selection, which realizes efficient solving of the approximate optimal solution of the minimum weighted vertex covering problem in a distributed environment without relying on centralized nodes and global information, and achieves efficient collaborative vertex covering only by using neighborhood information and local interaction.

[0007] To achieve the above-mentioned purposes of the application, the technical solutions of the present application are as follows:

[0008] The embodiment of the present application provides an efficient distributed vertex covering method based on Nash equilibrium selection, comprising:

[0009] The sum of vertex weights and the penalty for uncovered edges are used as indicators to establish a global optimization model of the minimum weighted vertex covering problem in a distributed network system;

[0010] The global optimization model is converted into a distributed collaborative decision-making model based on network ordinal potential game;

[0011] The neighbor set of each vertex in the distributed network system is determined and the action is initialized;

[0012] By communicating with their neighbors to obtain their action information, each vertex iterates and updates the actions of each vertex until the individual actions of all vertices no longer change, thus obtaining the optimal Nash equilibrium coverage scheme.

[0013] According to one aspect of the present invention, establishing a global optimization model for the minimum weighted vertex cover problem in a distributed network system, using the sum of vertex weights and the penalty for uncovered edges as indicators, includes:

[0014] Consider a distributed network system represented by a simple undirected connected graph Ξ=(V,E), where the set of points is V={v1,v2,...,v...}. n The edge set is E = {e} ij}, each vertex v i It has a positive weight ω i , i = 1, ..., n;

[0015] For a subset S of a vertex set V, if any edge in the undirected connected graph has at least one vertex belonging to S, then S is called a vertex cover.

[0016] For each vertex v i ∈V, its neighborhood is defined as Ω i ={v j |e ij ∈E}, its degree is d i =|Ω i |;

[0017] Given Ξ = (V, e) and The goal of the minimum weighted vertex cover problem is to find a vertex cover scheme with the minimum sum of weights. Let the subset S be represented as a 1×n binary vector a = (a1, a2, ..., a...). n ), where a i =1 means v i ∈S, a i =0 means

[0018] From a global optimization perspective, the objective of the minimum weighted vertex cover problem is to minimize the following global evaluation metric:

[0019]

[0020] Where λ is the introduced penalty factor for uncovered edges.

[0021] According to one aspect of the present invention, the step of transforming the global optimization model into a distributed collaborative decision-making model based on network ordinal potential game includes:

[0022] Establish a set of game participants Each vertex v in the undirected connected graph i ∈V as a game participant with communication and computational capabilities Then the set of game participants

[0023] Establish communication topology When vertex v i With v j When there are connecting edges between game participants, and There is a communication connection between them, that is, if e ij ∈E, then otherwise

[0024] Establish Action Set A i Each player in the game has the same set of actions A. i = {1, 0}, where a i =1 means v i ∈S, a i =0 means

[0025] Establish payment function u i , will u i Designed for game participants Zero neighbor count The piecewise function of the index is as follows:

[0026]

[0027] Wherein, the payment function u i Depends solely on game participants The neighborhood information, therefore u i (a) can also be written as in for Joint action by neighbors;

[0028] Establish a potential function φ. For any vertex covering scheme a, the potential function is defined as φ(a) = -f(a).

[0029] According to one aspect of the present invention, the step of determining the neighbor set of each vertex in a distributed network system and initializing the action includes:

[0030] Determine the neighbor set Ω for each game participant. i ={v j |e ij ∈E}, and initialize its action as Where t represents the number of iterations.

[0031] According to an aspect of the embodiment of the present application, the action information of the neighbors is obtained by each vertex communicating with the neighbors, and the action of each vertex is iteratively updated, comprising:

[0032] Each vertex communicates with the neighbors to obtain the action information of the neighbors, and performs individual local calculation, virtual regret value generation, evaluation factor generation and trial action generation, updates the action of each vertex, and iterates the process.

[0033] According to an aspect of the embodiment of the present application, the individual local calculation comprises:

[0034] Each game player performs local bidirectional communication with the neighbors to obtain the action information of the neighbors

[0035] The number of zero neighbors is calculated according to the following formula respectively The payment function value The best response And the regret value

[0036]

[0037]

[0038]

[0039]

[0040] Wherein, is the joint action of all game players except the game player in the tth coordination process;

[0041] Each game player sends the calculation result to each neighbor.

[0042] According to an aspect of the embodiment of the present application, the virtual regret value generation comprises:

[0043] Each game player determines the type of itself by using the zero neighbor information, if And The game player is a 0 action attractive player; if The game player is a 1 action attractive player; otherwise, the game player is a normal player;

[0044] The virtual regret value is generated according to the type of itself

[0045]

[0046] Each game participant sends the above result to each neighbor.

[0047] According to an aspect of the embodiment of the present application, the evaluation factor is generated, comprising:

[0048] Let All game participants Calculate the comparison factor for themselves and neighbors

[0049]

[0050] Wherein, α i is the maximum virtual regret value in, and sgn is a sign function;

[0051] According to the comparison factor, the evaluation factor is generated by using the limited greedy rule, if three conditions of and are simultaneously satisfied, then Otherwise

[0052] According to an aspect of the embodiment of the present application, the heuristic action generation, the action of each vertex is updated, comprising:

[0053] Each game participant generates a heuristic action according to the respective evaluation factor

[0054]

[0055] According to the type of the game participant, the action is updated according to the following formula,

[0056]

[0057] Wherein, rand({0, 1}) represents an action randomly selected from the action set {0, 1} with equal probability.

[0058] According to an aspect of the embodiment of the present application, if all vertex individuals are normal players, then the Nash equilibrium is a dominant Nash equilibrium; otherwise, the Nash equilibrium is a dominated Nash equilibrium.

[0059] Compared with the prior art, the present application has the following beneficial effects:

[0060] According to the scheme of the embodiment of the present application, each vertex in a network or an undirected connected graph is regarded as an independent game participant from the perspective of network ordinal potential game, a distributed collaborative decision-making model based on network ordinal potential game is constructed by establishing network communication topology, individual strategy set, local payment function and potential function, game participants (0 action attractive player, 1 action attractive player and normal player) and Nash equilibrium (dominant Nash equilibrium and inferior Nash equilibrium) are classified by using local neighborhood information, and game learning and refinement algorithms are designed, thereby providing a more efficient solution for vertex covering in a distributed environment.

[0061] The method is a pure distributed algorithm, each vertex only needs to use neighborhood communication and local rules to update action, without the intervention of a centralized node, and can realize autonomous collaborative decision-making without relying on a centralized control node and global information, has the advantages of fast response speed and strong robustness, and is suitable for autonomous solving of a minimum weighted vertex covering scheme for a large-scale network system.

[0062] The method uses optimal response and random selection to design a distributed synchronous iterative optimization algorithm, and can realize self-organizing collaborative decision-making and intelligent emergence based on neighborhood interaction on a complex network. Through the characteristic analysis of Nash equilibrium and the classification based on neighborhood information, the emergence of inefficient Nash equilibrium is effectively identified and prevented. Through learning and improvement of efficient Nash equilibrium, the emergence of global efficient collaboration scheme is realized, which is helpful to improve the collaboration efficiency at the system level.

[0063] The method can be transplanted to other distributed collaborative optimization problems that can be expressed as ordinal potential game, such as distributed multi-star collaborative task allocation and multi-agent set covering. BRIEF DESCRIPTION OF DRAWINGS

[0064] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed in the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0065] Figure 1 An embodiment of the present application discloses a high-efficiency distributed vertex covering method based on Nash equilibrium selection, and a flowchart of the embodiment is shown in the figure;

[0066] Figure 2 An embodiment of the present application discloses a distributed solving algorithm based on network ordinal potential game, and a flowchart of the embodiment is shown in the figure;

[0067] Figure 3 An embodiment of the present application discloses a non-vertex covering example, and a flowchart of the embodiment is shown in the figure;

[0068] Figure 4 An example of a vertex cover disclosed by embodiments of the application is schematically represented;

[0069] Figure 5 An example of a minimum-weight vertex cover disclosed by embodiments of the application is schematically represented;

[0070] Figure 6 An example of a dominated Nash equilibrium disclosed by embodiments of the application is schematically represented;

[0071] Figure 7 An example of a dominated Nash equilibrium disclosed by embodiments of the application is schematically represented;

[0072] Figure 8 A convergence curve of the global evaluation function f disclosed by embodiments of the application is schematically represented;

[0073] Figure 9 A convergence curve of the number of covered vertices |S| disclosed by embodiments of the application is schematically represented;

[0074] Figure 10 A convergence curve of the number of non-normal individuals disclosed by embodiments of the application is schematically represented. DETAILED DESCRIPTION

[0075] The description of the embodiments of the specification should be considered in connection with the accompanying drawings, which are incorporated in part herein. In the drawings, the proportions of the embodiments can be exaggerated for the sake of convenience or clarity and the same reference numbers can be used in different drawings to identify the same or similar elements. Also, parts of structures that are not necessary for describing the present application can not be shown or described.

[0076] The description of the embodiments herein should not be interpreted as implying any particular order of the steps of the described embodiments. The described embodiments can be implemented in any order or simultaneously. Any reference signs in the claims should not be construed as limiting the scope of the claims. The description herein is intended to be illustrative, and not to limit the scope of the application. Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the disclosure. Accordingly, the scope of the application will not be limited by these specific illustrative embodiments.

[0077] This invention discloses an efficient distributed vertex cover method based on Nash equilibrium selection, proposed using Nash equilibrium learning and autonomous selection theory in game theory. According to the concept of this invention, the method first constructs a global mathematical model of the minimum weighted vertex cover problem in a distributed environment, then transforms it into an ordinal potential game model on a communication network. Subsequently, it sequentially goes through steps such as action initialization, individual local computation, virtual regret value generation, evaluation factor generation, and action update. Finally, it jumps back to the individual local computation step for iterative iteration until final convergence, i.e., the distributed network system stabilizes, and outputs the distributed optimization result (i.e., the approximate optimal solution of the minimum weighted vertex cover), achieving a distributed and efficient solution to the minimum weighted vertex cover problem.

[0078] like Figure 1 As shown in the embodiments of the present invention, the above-mentioned efficient distributed vertex covering method based on Nash equilibrium selection specifically includes the following steps: S110, establishing a centralized global optimization model for the minimum weighted vertex covering problem in the distributed network system using the sum of vertex weights and the penalty for uncovered edges as indicators; S120, transforming the global optimization model into a distributed collaborative decision-making model based on network ordinal potential game; S130, determining the neighbor set of each vertex in the distributed network system and initializing actions; S140, obtaining the action information of each vertex through communication with its neighbors, and iteratively updating the actions of each vertex until the individual actions of all vertices no longer change. At this point, the coordination (iteration) process ends, and an optimized Nash equilibrium covering scheme is obtained. Through the above scheme, an efficient solution to the approximate optimal solution of the minimum weighted vertex covering problem can be achieved in a distributed environment without relying on centralized nodes and global information, achieving efficient collaborative vertex covering only by utilizing neighborhood information and local interactions.

[0079] In one embodiment, the specific process of establishing a global optimization model for the minimum weighted vertex cover problem in a distributed network system in step S110, using the sum of vertex weights and the penalty for uncovered edges as indicators, includes the following steps: Consider a distributed network system represented by a simple undirected connected graph Ξ=(V,E), where the vertex set is V={v1,v2,...,v...} n The edge set is E = {e} ij}, each vertex v i It has a positive weight ω i Let i = 1, ..., n; for a subset S of a vertex set V, if every edge in an undirected connected graph has at least one vertex belonging to S, then S is called a vertex cover. For each vertex v i ∈V, its neighborhood is defined as Ω i ={v j |e ij ∈E}, its degree is d i =|Ωi |. Given Ξ=(V,E) and The goal of the minimum weighted vertex cover problem is to find a vertex cover scheme with the minimum sum of weights. Let the subset S be represented as a 1×n binary vector a = (a1, a2, ..., a...). n ), where a i =1 means v i ∈S, a i =0 means From a global optimization perspective, the objective of the minimum weighted vertex cover problem is to minimize the following global evaluation metric:

[0080]

[0081] Where λ is the introduced penalty factor for uncovered edges.

[0082] In one embodiment, the specific process of transforming the global optimization model into a distributed collaborative decision-making model based on network ordinal potential game in step S120 includes the following steps: S121, establishing a set of game participants. Each vertex v in the undirected connected graph i ∈V as a game participant with communication and computational capabilities (possessing autonomy). Then the set of game participants S122. Establish communication topology When vertex v i With v j When there are connecting edges between game participants, and There is a communication connection between them, that is, if e ij ∈E, then otherwise S123, Establish Action Set A i Each player in the game has the same set of actions A. i = {1, 0}, where a i =1 means v i ∈S, a i =0 means S124. Establish the payment function u i To introduce heuristic information, u i Designed for game participants The number of zero neighbors (zero neighbors are neighbors with a strategy of zero). The piecewise function of the indicator, the payment function u i As shown in the following formula:

[0083]

[0084] As can be seen, the payment function ui only rely on the neighborhood information of game participants , thus u i (a) can also be written as where is the joint action of neighbors. S125, a potential function φ also needs to be established, which is defined as φ(a) = -f(a) for any vertex cover scheme a.

[0085] Next, the minimum weighted vertex cover problem in the above distributed network system is solved using a distributed solution process based on network ordinal potential game as shown in Figure 2 . Figure 2 Y in the above formula indicates that the condition is met (Yes), and N indicates that the condition is not met (No).

[0086] In one embodiment, as shown in Figure 2 , the specific process of determining the neighbor set of each vertex in the distributed network system and initializing the action in step S130 includes: determining the neighbor set Ω i = {v j |e ij ∈E} of each game participant, and initializing the action of each game participant as where t represents the iteration number.

[0087] In one embodiment, as shown in Figure 2 , the specific process of obtaining the action information of neighbors by each vertex communicating with neighbors and iteratively updating the action of each vertex in step S140 includes: each vertex communicates with neighbors to obtain neighbor action information, and performs individual local calculation, virtual regret value generation, evaluation factor generation, and trial action generation, updates the action of each vertex, and iteratively loops this process.

[0088] Specifically, the specific process of individual local calculation includes the following steps: in the tth coordination (iteration) process, each game participant performs local bidirectional communication with neighbors to obtain neighbor action information Then, the zero neighbor number , the payment function value , the best response , and the regret value are calculated according to the following formulas respectively.

[0089]

[0090]

[0091]

[0092]

[0093] wherein, is the joint action of all game players except game player in the tth coordination process. Each game player sends the above calculation result to each neighbor.

[0094] Specifically, the process of generating the virtual regret value includes the following steps: each game player determines its own type by using zero-neighbor information, if and then the game player is a 0-attracting player; if then the game player is a 1-attracting player; otherwise, the game player is a normal player. Then, the game player generates a virtual regret value

[0095]

[0096] Finally, each game player sends the above result to each neighbor.

[0097] Specifically, the process of generating the evaluation factor includes the following steps: let each game player calculate a comparison factor

[0098]

[0099] wherein, α i is the maximum virtual regret value in, and sgn is a sign function. Then, according to the comparison factor, the game player generates an evaluation factor by using a limited greedy rule, if three conditions (i)

[0100] (ii) (iii) are simultaneously satisfied, then otherwise

[0101] Specifically, the process of generating the trial action and updating the action of each vertex includes the following steps: each game player generates a trial action according to its own evaluation factor as follows:

[0102]

[0103] According to the type of the game player, the game player updates the action according to the following formula,

[0104]

[0105] where rand({0, 1}) denotes an action randomly selected from the action set {0, 1} with equal probability.

[0106] In the above steps, according to whether all vertex individuals are normal players, the Nash equilibrium in the game process can be classified as follows: if all vertex individuals are normal players, the Nash equilibrium is a dominant Nash equilibrium; otherwise, the Nash equilibrium is a dominated Nash equilibrium. Figure 2 The distributed solution process in the above formula is iterated by loop until converging to a dominant Nash equilibrium solution.

[0107] In this embodiment, the experimental object is a -Rényi random distributed network, where the weight ω of each vertex is i is randomly generated and normalized between 0 and 1. The experimental environment is MATLAB 2016a version, the operating system is Win10, 3.07Ghz, and the memory is 4G.

[0108] Step 1: Establish a global optimization model of minimum weighted vertex cover

[0109] For a -Rényi random network with n = 5000 vertices and |E| = 124459 edges, the global minimization objective function is as follows:

[0110]

[0111] where λ = 3.9751 × 10 -4 .

[0112] Step 2: Establish a distributed collaborative decision-making model based on network ordinal potential game

[0113] Each vertex in the graph Ξ is regarded as a game player, and a network ordinal potential game model is established as follows:

[0114] The set of game players Each vertex in the graph is regarded as a game player The set of game players

[0115] Communication topology If vertex v i has a connection edge with v j , then and are communicatively connected, i.e., if e ij ∈E, then Otherwise,

[0116] Action Set A i : Each game player has the same action set A i = {1,0}, where a i = 1 means v i ∈ S, a i = 0 otherwise;

[0117] Payment Function u i : For any joint action a, the payment function u i is designed as a piecewise function as follows:

[0118]

[0119] Potential Function φ: For any covering scheme a, the potential function is defined as φ(a) = -f(a).

[0120] Step Three: Action Initialization

[0121] For any vertex v i ∈ V, determine the neighbor set Ω i = {v j | e ij ∈ E} and initialize its action as

[0122] Step Four: Individual Local Computation

[0123] In the t-th coordination, each performs the following local computation and sends the result to its neighbors:

[0124] Using the neighbor action information According to the computation of zero neighbors Payment function value Best response And regret value

[0125]

[0126]

[0127]

[0128]

[0129] Step Five: Virtual Regret Value Generation

[0130] Each game player determines its own type: if and then the game player is a 0-action attractive player; if Then If a player is attracted to the game; otherwise, the game player is a normal player.

[0131] Each game player generates a virtual regret value according to the type of the player

[0132]

[0133] Each game player sends the above result to each neighbor.

[0134] Step six: evaluation factor generation

[0135] Each participant calculates the comparison factor for all individuals in

[0136]

[0137] Each participant generates the evaluation factor by using the limited greedy rule: if (i) (ii) (iii) if the three conditions of i < j < k are met, then otherwise,

[0138] Step seven: action update

[0139] Each participant generates a trial action according to the following rule and updates the action:

[0140]

[0141]

[0142] Step eight: loop iteration

[0143] The above process jumps back to iteration step four until all individual actions no longer change, and the coordination process ends, and the current action combination is the approximate optimal result of the minimum weighted vertex cover problem.

[0144] Figures 3-7 The illustration of different concepts in the method is given, a i = 1 indicates that the vertex belongs to S, a i = 0 indicates that the vertex does not belong to S, A, B, C, D, E, F are vertex names, and the numbers on the right side are the weight values of the vertices. Among them, Figure 3 ​​For non-vertex cover example, Figure 4 For vertex cover example, Figure 5 For minimum weighted vertex cover example, Figure 6 For inferior Nash equilibrium example, Figure 7 For superior Nash equilibrium example. Figure 8 、 Figure 9 With Figure 10 The convergence curve of the distributed optimization obtained by the method is shown in the figure, wherein, Figure 8 The convergence curve of the global evaluation index f is shown in the figure, Figure 9 The convergence curve of the number of covered vertices |S| (indicating the number of elements in the alternative point set S) is shown in the figure, Figure 10 The convergence curve of the number of abnormal individuals γ is shown in the figure. The results show that the distributed minimum weighted vertex cover algorithm proposed by the method can avoid falling into the low-efficiency solution represented by the inferior Nash equilibrium without relying on the centralized control vertex and global information, and can guarantee the global convergence of the superior Nash equilibrium solution, and obtain a cover scheme close to the global optimum.

[0145] In summary, the present application designs a distributed optimization method for the minimum weighted vertex cover problem in the distributed environment by using game learning and refinement theory. Each vertex in the undirected connected graph is regarded as a game participant with autonomous ability, and the original problem is converted into a sequential potential game Nash equilibrium solving and autonomous selection problem on the corresponding communication network by constructing the communication topology, action set, payment function and potential function. The classification method of game participants and Nash equilibrium is given by using local information, and a synchronous iterative learning algorithm based on limited greedy best response and random selection mechanism is designed, which can effectively guarantee the finite time convergence of the superior Nash equilibrium solution. Based on the above reasons, the present application gets rid of the dependence on the centralized control node and global information, destroys the stability of the low-efficiency inferior Nash equilibrium, realizes the rapid solution of the superior Nash equilibrium, and can greatly improve the global performance of the cover scheme and reduce the negotiation and calculation time. In addition, the present application can not only be used to solve the distributed minimum weighted vertex cover, but also can be transplanted to other distributed optimization problem solving, such as set cover, cooperative task allocation, etc.

[0146] The serial numbers of the above-mentioned steps involved in the method of the present application do not mean the order of the execution of the method, and the execution order of each step should be determined according to its function and internal logic, and should not constitute any limitation on the implementation process of the embodiment of the present application.

[0147] The above only describes the preferred embodiments of the present application and should not be used to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. An efficient distributed vertex cover method based on Nash equilibrium selection, comprising: establishing a global optimization model of a minimum weighted vertex cover problem in a distributed network system with the sum of vertex weights and the penalty for uncovered edges as indicators; transforming the global optimization model into a distributed collaborative decision-making model based on network ordinal potential game; determining the neighbor set of each vertex in the distributed network system and initializing the action; obtaining neighbor action information through communication between each vertex and its neighbors, and iteratively updating the action of each vertex until the individual action of all vertices no longer changes, obtaining an optimized Nash equilibrium cover scheme; the transformation of the global optimization model into a distributed collaborative decision-making model based on network ordinal potential game comprises: establishing a set of game players , a set of game players , a set of game players , a set of game players , a set of game players ; wherein, , a set of game players , a set of game players Establishing a communication topology When there is a connecting edge between a vertex and , there is a communication connection between the game participants and , then ; otherwise ; establishing a set of actions each game player has the same set of actions wherein denotes , denotes ; wherein, denotes a subset of a vertex cover of the point set ; Establishing a payment function , the designed to game participants the number of zero neighbors as an index of piecewise function, as follows: in, Represents vertices Degree, payment function Depends solely on game participants Neighborhood information, for Joint action by neighbors As vertices The neighborhood, As vertices The weights; Establishing the potential function For any vertex cover scheme , the potential function is defined as , is the global evaluation index; the determination of the neighbor set of each vertex in the distributed network system and the initialization of the action comprises: determining a set of neighbors for each game player and initializing their actions to where denotes the number of iterations; updating the action of each vertex comprises: each vertex communicates with its neighbors to obtain its action information, calculates the zero neighbor number, the local payment value, and the best response regret value, and sends the calculation results to each neighbor; each vertex judges its type using its zero neighbor information, generates a virtual regret value using the regret value, and sends its type and the generated virtual regret value to each neighbor; each vertex calculates a comparison factor for itself and its neighbors, generates an evaluation factor using the comparison factor, generates a trial action according to the evaluation factor, and updates the current action using the trial action.

2. The method of claim 1, wherein, the establishment of the global optimization model of the minimum weighted vertex cover problem in the distributed network system with the sum of vertex weights and the penalty for uncovered edges as indicators comprises: Consider a distributed network system represented by a simple undirected connected graph where the set of points is , the set of edges is , each vertex has a positive weight , ; a subset of a point set a subset of a point set a subset of a point set a subset of a point set a subset of a point set For each vertex , define its neighborhood as , with degree ; Given and , the goal of the minimum-weight vertex cover problem is to find a vertex cover scheme with the minimum sum of weights, denoting a subset as a binary vector , where , and denotes , denotes ; from the perspective of global optimization, the objective of the minimum weighted vertex cover problem is to minimize the following global evaluation indicator: wherein, is an introduced uncovered edge penalty factor.

3. The method of claim 2, wherein, the obtaining of neighbor action information through communication between each vertex and its neighbors and the iterative updating of the action of each vertex comprises: each vertex communicates with its neighbors to obtain neighbor action information, and performs individual local calculation, virtual regret value generation, evaluation factor generation, and trial action generation, updates the action of each vertex, and iteratively cycles this process.

4. The method of claim 3, wherein, the individual local calculation comprises: Each game participant communicates with neighbors for local bidirectional communication to obtain neighbor action information ; The zero neighbors number is calculated according to the following formula respectively , the payment function value , the best response and the regret value : wherein, is the first coordination process joint action of all game players except the game player each game participant sends the above calculation results to each neighbor.

5. The method of claim 4, wherein, the virtual regret value generation comprises: Each game player uses zero neighbor information to determine its own type, if and , then the game player is a 0 action attractor player; if , then the game player is a 1 action attractor player; otherwise the game player is a normal player; Generating virtual regret values according to own type : each game participant sends the above results to each neighbor.

6. The method of claim 5, wherein, the evaluation factor generation comprises: Let , all game players , compute comparison factors for themselves and their neighbors , wherein is the maximum virtual regret value in is the sign function; According to the comparison factor, an evaluation factor is generated using a limited greedy rule , if three conditions are met simultaneously ; ; and , then ; otherwise , wherein is a comparison factor of a game participant itself, is a comparison factor of a game participant and a neighbor .

7. The method of claim 6, wherein, the trial action generation, which updates the action of each vertex, comprises: Each game participant generates a tentative action based on respective evaluation factors , updating the action according to the type of the game participant itself according to the following formula, wherein, represents an action from the set of actions an action chosen randomly with equal probability.

8. The method of claim 7, wherein, if all vertices are normal players, the Nash equilibrium is a dominant Nash equilibrium; otherwise, the Nash equilibrium is a dominated Nash equilibrium.

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