Aggregating local behaviors based upon a discrete ... - IEEE Xplore
Aggregating Local Behaviors Based upon a Discrete Lagrange Multiplier Method Yi Tang Department of Information Science Guangzhou University Guangzhou, 510405, China [email protected] State Key Lab of Information Security Institute of Software, CAS Beijing, 100039, China
Abstract When solving a distributed problem based on a multiagent system, the local behaviors of agents will be aggregated to the global behaviors of the multi-agent system towards a solution state. This paper presents a distributed discrete Lagrange multiplier (DDLM) method for solving distributed constraint satisfaction problems (distributed CSPs). In this method, the local behaviors of agents are aggregated as a descent direction of an objective function corresponding to the problem at hand. Thus, a trend to a solution state will be formed. Furthermore, we provide three techniques to speed up the aggregation of agents’ local behaviors. Through experiments on benchmark graph coloring problems, we validate the effectiveness of the presented DDLM method as well as the three techniques in solving distributed CSPs.
the variables in . To find a solution to a distributed CSP requires agents to cooperate so as to find an assignment of variables that satisfies all constraints. The graph coloring problem (GCP) is an important class with vertex set and of CSPs. Given a graph edge set , and an integer , a of is a mapping : where , . The GCP is to examine whether or not a -coloring of exists. This paper proposes a distributed discrete Lagrange multiplier (DDLM) method for distributed CSP solving. We first present the theoretical framework of the DDLM method. In this framework, we transform a CSP to a minimization problem and employ a multi-agent system to determine its descend direction in a distributed manner. We further propose three techniques, i.e., normal neighborhood, enlarged neighborhood, and cutoff strategy, to improve the performance of the DDLM method.
2 The DDLM Method for Distributed CSPs
1 Introduction When solving a distributed problem based on a multiagent system, an agent tries to solve a subproblem associated with the variables it represents [3, 7, 8]. Specifically, an agent locally behaves to assign values to its variables. In doing so, it needs to coordinate with other related agents, usually called neighboring agents, so as to make sure that the values of their variables are consistent. Through coordination, the local behaviors of agents are aggregated to the global behaviors of the multi-agents system to find a solution to the problem to be solved. A distributed constraint satisfaction problem (distributed , where is a set of CSP) is a quaternion variables, is a set of domains corresponding to , is a set of constraints, and is a set of agents used to represent
Jiming Liu and Xiaolong Jin Department of Computer Science Hong Kong Baptist University Kowloon Tong, Hong Kong jiming, jxl@comp.hkbu.edu.hk
In this section, we will present the theoretical framework of the distributed discrete Lagrange multiplier (DDLM) method.
2.1 The Theoretical Framework
Given a distributed CSP where , , , and is the domain of the variable in , we assume that all the variables are divided into different non-empty groups. Assume the -th group ½ ¾ where denotes the number of variables in this group, we denote Ü ½ ܾ Ü as the point Ü ½ ½ ¾ , where Ü ¾ . Usually, a distributed CSP can be transformed into the
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
following equality optimization problem:
Ü Ü ¼
(1)
where Ü is a global objective function that reflects the current solving status and Ü = ( Ü , Ü , ...,
Ü ) is a vector of equality constraints. The corresponding Lagrange function is as follows:
Ü Ü (2) where . is called a Lagrange mul-
tiplier. The of two points Ü: (Ü , Ü , ..., Ü ) and Ý: (Ý , Ý , ..., Ý ) is , if there exists different scalar quan. tities between them. We denote it as Ü Ý The set Ü Ý Ü Ý is the neighborhood of the Ü. For the problem (1), the point Ü is , ¼ and the feasible point Ü is a if Ü if Ü Ü for any feasible point Ü Ü . Now, we introduce the notion of [5]. If Ü satisfies Ü Ü Ü for all sufficiently close to and for all Ü Ü , it is a saddle point of Ü . Note that a point Ü is a local minimum of problem (1), if (Ü , ) is a saddle point of the corresponding Lagrange function for some [5]. In order to find a saddle point from any random initial point, we will perform the following iterative operations:
Ü
(3) Ü Ü
Ü (4) where denotes the -th iteration, Ü is a decreasing
direction of equation (2) and
is the step length vector in -th iteration. If we keeps the step length vector nonnegative in every iteration, we have the following theorem.
Theorem 1 [5] A saddle point (Ü , ) of Ü is reached iff the iteration (3)(4) terminates at Ü .
2.2 Transformation of Distributed GCPs
Considering the -th edge in a graph with vertices and edges, the vertices and can be viewed as the variables and , respectively, where . We define the function:
where Ü . Constructing the equality op Ü
timization problem:
Ü
Ü
We can formulate the corresponding Lagrange function as follows:
Ü
Ü
(5)
¼
Ü
Ü
(6)
As discussed previously, the saddle point Ü of equation (6) implies that Ü ¼. It means Ü is a solution to problem (5). Given the above transformation, we have found a way to find a solution to a distributed GCP. Specifically, we need to aggregate the local behaviors of agents to minimize the value of equation (6). We call this method a distributed discrete Lagrange multiplier (DDLM) method. Next, we need to appropriately define the local behaviors of agents so that a solution to the optimization problem can be found quickly.
3 Applying the DDLM Method to Distributed GCPs In this section, we will give three simple techniques based on the DDLM method to define the local behaviors of agents. We assume that each agent represents one variable and its step length in each cycle is one. We also assume that there is a cyclic schedule such that all agents can finish their local decisions in a cycle. Two neighboring agents cannot change the values of their variables at the same time, i.e., the colors of two neighboring vertices cannot be changed simultaneously.
3.1 Normal Neighborhood and Central Agents Suppose two variables and are represented by agents and , respectively. is a agent of if edge . All neighboring agents of construct the !" . Let !" !" . We denote the identity code of agent as and the maximal value decrease of function (6) when changes the value of the variable in the -th cycle as .
Definition 1 (central agent) Agent is called a agent in the -th cycle if it satisfies the following conditions:
; and .
We define the local behaviors of agent in the -th cycle as follows: 1. Exchange the value information of represented variable with its neighbors;
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
C
A
Definition 3 (subcentral agent) Agent is called a agent in the -th cycle if it is not central and satisfies the following conditions for :
B
D
E
¯
F
¯
Figure 1. A GCP example
Now we introduce the enlarged neighborhood :
and send it to all its neighbors; Decide if it can become based on all the other received from its neighbors; If agent is , it will do the following: (a) If , changes the value of such
2. Compute 3. 4.
¼
that it can minimize the equation (6);
(b) If and there exists unsatisfied constraints associated with its represented variable, increases the Lagrange multipliers by one for all unsatisfied edges and sends them to associated neighbors.
This method is similar to the DB method [8]. According to definition 1, if and are central agents, and are not neighbors. Since only central agents can change the values of their variables, the behavior of each central agent cannot directly affect other central agents in a cycle. Thus, in each cycle, the behaviors of central agents are aggregated to a global behavior to optimize equation (6). When a central agent cannot optimize equation (6) by changing the value of its variable and there exist unsatisfied constraints associated with its variable, it means that the central agent gets stuck in a local minimum. Then, an update on the Lagrange multipliers associated with the unsatisfied constraints will help the agent escape from the local minimum. However, there exist some problems in this method. Consider the 3-colorable problem in Figure 1. Assume agents represent vertices , respectively and all vertices have the same color initially. We need three cycles to find a solution by the above strategy. Note that the behavior of the central agent cannot directly influences agent . If can change its color simultaneously, a solution can be found with two cycles. As motivated by this observation, we introduce the notion of enlarged neighborhood to improve the performance.
3.2 Enlarged Neighborhood and Subcentral Agents Definition 2 (frozen agent) Agent is frozen in the -th cycle iff that is central.
, is frozen; if , either or is frozen; .
¯ if
such
¼ ¾
¼
Thus, for each agent , we can redefine its local behaviors in the -th cycle as follows: 1. Exchange the value information of its represented variable to its neighbors; 2. Compute bors;
and send it to its enlarged neigh-
If agent is , it will perform the following behaviors ¼
from its enlarged neighbors 3. Receive all the agent in this cycle; and decide if it can become a
4.
to change its local environment:
- message (the content is that ) to all agents in . , change the value of its repre-
(a) Send a agent is
(b) If sented variables;
and there exist unsatisfied con(c) If straints associated with its represented variable, increase the Lagrange multipliers by one for all unsatisfied edges and send them to associated neighbors.
- messages and decide if it can become agent in this cycle; If agent is , change the value of its represented variable when .
5. Receive a 6.
3.3 Cutoff Strategy A potential problem of equations (3) and (4) lies in the unlimited increase in some Lagrange multipliers. This may make the multi-agent system much difficult to escape from a local optimum. Many methods have been proposed to keep relatively even multipliers for centralized systems [4, 5, 6]. A rapid restart strategy [2] was introduced to eliminate the heavy-tail behavior of a backtrack procedure for propositional satisfiability and constraint satisfaction problems. In order to make the multipliers relatively even in distributed GCP solving, we introduce a cutoff threshold for Lagrange multipliers. The theshold is a value of cycles that triggers the system to reset Lagrange multipliers to zero. We will apply this strategy to the global control schedule.
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
Table 1. A comparison between DDLM DDLM . Testset Miles
REG (mulsol) REG (zeroin)
Instance miles500 miles750 miles1000 mulsol.i.1 mulsol.i.2 mulsol.i.3 zeroin.i.1 zeroin.i.2 zeroin.i.3
and
DDLM 212 1,082 1,684 1,645 1,910 1,988 3,046 1,986 2,022
DDLM 233 1,153 1,722 1,738 1,997 2,089 3,294 2,049 2,028
Table 2. A comparison among DDLM DDLM , and DDLM . Instance queen6 6 queen7 7 queen8 8 queen9 9
DDLM 2,008 1,724 3,874 2,552 7,688 4,661 11,943 6,519
DDLM 2,860 1,055 15,564 3,404 9,386 5,232 10,089 7,129
5 Conclusion
,
DDLM 1,642 1,080 2,043 1,631 5,391 3,788 7,801 5,727
4 Experiments and Discussions We conducted our experiments on feasible benchmark GCP instances: Miles graphs, REG graphs, and Queen graphs [1]. We ran each instance 100 times with random initial assignments. Initially, we set Lagrange multipliers to zero. Table 1 shows the average numbers of cycles needed to find a solution by two DDLM methods with normal neighborhood (DDLM ) and enlarged neighborhood (DDLM ), respectively. As compared to DDLM , DDLM needs less cycles. In DDLM , each agent sends its to its enlarged neighborhood. If an agent becomes central in a certain cycle, it also sends to its enlarged neighborhood. This gives the agents in enlarged neighborhood a chance to consider their local behaviors in a more global level. Table 2 shows the experimental results of the DDLM methods with normal neighborhood (DDLM ), enlarged neighborhood (DDLM ), and cutoff value (DDLM ) on four Queen graph instances. In the experiment, we set the cutoff threshold to ¾¼¼. In Table 2, the first row corre-
sponding to each each instance shows the mean cycles and the second row shows the median cycles. We find that the performance in solving these instances depends on the initial assignment. This point is shown by the larger differences between the corresponding mean and median values. We further find that the cutoff strategy significantly reduces the number of cycles to find a solution in those instances. Since the strategy is to periodically reset Lagrange multipliers, it shows that keeping relatively even multipliers is a helpful for the performance of the DDLM methods.
We have presented a distributed discrete Lagrange multiplier (DDLM) method for solving distributed CSPs. According to this method, the local behaviors of agents are aggregated to find a global descent direction of an objective function corresponding to the CSP to be solved. Through experiments, we have validated the effectiveness of the DDLM method. We have improved its performance by introducing three techniques, i.e., normal neighborhood, enlarged neighborhood, and cutoff strategy.
References [1] http://mat.gsia.cmu.edu/COLORING03. [2] C. P. Gomes, B. Selman, N. Crato, and H. Kautz. Heavytailed phenomena in satisfiability and constraint satisfaction problems. Journal of Automated reasoning, 24:67–100, 2000. [3] J. Liu, H. Jing, and Y. Y. Tang. Multi-agent oriented constraint satisfaction. Artificial Intelligence, 138:101–144, 2002. [4] D. Schuurmans and F. Southey. Local search characteristics of incomplete SAT procedures. Artificial Intelligence, 132(2):121–150, 2001. [5] Y. Shang and B. W. Wah. A discrete lagrangian-based globalsearch method for solving satisfiability problems. Journal of global optimization, 12(1):61–99, 1998. [6] Z. Wu and B. W. Wah. An efficient global-search strategy in discrete lagrangian methods for solving hard satisfiability problems. In Proceedings of AAAI/IAAI 2000, pages 310– 315, 2000. [7] M. Yokoo, E. H. Durfee, T. Ishida, and K. Kuwabara. The distributed constraint satisfaction problem: Formalization and algorithms. IEEE Transaction on Knowledge and Data Engineering, 10(5):673–685, 1998. [8] M. Yokoo and K. Hirayama. Distributed breakout algorithm for solving distributed constraint satisfaction problems. In Proceedings of ICMAS96, pages 401–408, 1996.
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
Abstract When solving a distributed problem based on a multiagent system, the local behaviors of agents will be aggregated to the global behaviors of the multi-agent system towards a solution state. This paper presents a distributed discrete Lagrange multiplier (DDLM) method for solving distributed constraint satisfaction problems (distributed CSPs). In this method, the local behaviors of agents are aggregated as a descent direction of an objective function corresponding to the problem at hand. Thus, a trend to a solution state will be formed. Furthermore, we provide three techniques to speed up the aggregation of agents’ local behaviors. Through experiments on benchmark graph coloring problems, we validate the effectiveness of the presented DDLM method as well as the three techniques in solving distributed CSPs.
the variables in . To find a solution to a distributed CSP requires agents to cooperate so as to find an assignment of variables that satisfies all constraints. The graph coloring problem (GCP) is an important class with vertex set and of CSPs. Given a graph edge set , and an integer , a of is a mapping : where , . The GCP is to examine whether or not a -coloring of exists. This paper proposes a distributed discrete Lagrange multiplier (DDLM) method for distributed CSP solving. We first present the theoretical framework of the DDLM method. In this framework, we transform a CSP to a minimization problem and employ a multi-agent system to determine its descend direction in a distributed manner. We further propose three techniques, i.e., normal neighborhood, enlarged neighborhood, and cutoff strategy, to improve the performance of the DDLM method.
2 The DDLM Method for Distributed CSPs
1 Introduction When solving a distributed problem based on a multiagent system, an agent tries to solve a subproblem associated with the variables it represents [3, 7, 8]. Specifically, an agent locally behaves to assign values to its variables. In doing so, it needs to coordinate with other related agents, usually called neighboring agents, so as to make sure that the values of their variables are consistent. Through coordination, the local behaviors of agents are aggregated to the global behaviors of the multi-agents system to find a solution to the problem to be solved. A distributed constraint satisfaction problem (distributed , where is a set of CSP) is a quaternion variables, is a set of domains corresponding to , is a set of constraints, and is a set of agents used to represent
Jiming Liu and Xiaolong Jin Department of Computer Science Hong Kong Baptist University Kowloon Tong, Hong Kong jiming, jxl@comp.hkbu.edu.hk
In this section, we will present the theoretical framework of the distributed discrete Lagrange multiplier (DDLM) method.
2.1 The Theoretical Framework
Given a distributed CSP where , , , and is the domain of the variable in , we assume that all the variables are divided into different non-empty groups. Assume the -th group ½ ¾ where denotes the number of variables in this group, we denote Ü ½ ܾ Ü as the point Ü ½ ½ ¾ , where Ü ¾ . Usually, a distributed CSP can be transformed into the
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
following equality optimization problem:
Ü Ü ¼
(1)
where Ü is a global objective function that reflects the current solving status and Ü = ( Ü , Ü , ...,
Ü ) is a vector of equality constraints. The corresponding Lagrange function is as follows:
Ü Ü (2) where . is called a Lagrange mul-
tiplier. The of two points Ü: (Ü , Ü , ..., Ü ) and Ý: (Ý , Ý , ..., Ý ) is , if there exists different scalar quan. tities between them. We denote it as Ü Ý The set Ü Ý Ü Ý is the neighborhood of the Ü. For the problem (1), the point Ü is , ¼ and the feasible point Ü is a if Ü if Ü Ü for any feasible point Ü Ü . Now, we introduce the notion of [5]. If Ü satisfies Ü Ü Ü for all sufficiently close to and for all Ü Ü , it is a saddle point of Ü . Note that a point Ü is a local minimum of problem (1), if (Ü , ) is a saddle point of the corresponding Lagrange function for some [5]. In order to find a saddle point from any random initial point, we will perform the following iterative operations:
Ü
(3) Ü Ü
Ü (4) where denotes the -th iteration, Ü is a decreasing
direction of equation (2) and
is the step length vector in -th iteration. If we keeps the step length vector nonnegative in every iteration, we have the following theorem.
Theorem 1 [5] A saddle point (Ü , ) of Ü is reached iff the iteration (3)(4) terminates at Ü .
2.2 Transformation of Distributed GCPs
Considering the -th edge in a graph with vertices and edges, the vertices and can be viewed as the variables and , respectively, where . We define the function:
where Ü . Constructing the equality op Ü
timization problem:
Ü
Ü
We can formulate the corresponding Lagrange function as follows:
Ü
Ü
(5)
¼
Ü
Ü
(6)
As discussed previously, the saddle point Ü of equation (6) implies that Ü ¼. It means Ü is a solution to problem (5). Given the above transformation, we have found a way to find a solution to a distributed GCP. Specifically, we need to aggregate the local behaviors of agents to minimize the value of equation (6). We call this method a distributed discrete Lagrange multiplier (DDLM) method. Next, we need to appropriately define the local behaviors of agents so that a solution to the optimization problem can be found quickly.
3 Applying the DDLM Method to Distributed GCPs In this section, we will give three simple techniques based on the DDLM method to define the local behaviors of agents. We assume that each agent represents one variable and its step length in each cycle is one. We also assume that there is a cyclic schedule such that all agents can finish their local decisions in a cycle. Two neighboring agents cannot change the values of their variables at the same time, i.e., the colors of two neighboring vertices cannot be changed simultaneously.
3.1 Normal Neighborhood and Central Agents Suppose two variables and are represented by agents and , respectively. is a agent of if edge . All neighboring agents of construct the !" . Let !" !" . We denote the identity code of agent as and the maximal value decrease of function (6) when changes the value of the variable in the -th cycle as .
Definition 1 (central agent) Agent is called a agent in the -th cycle if it satisfies the following conditions:
; and .
We define the local behaviors of agent in the -th cycle as follows: 1. Exchange the value information of represented variable with its neighbors;
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
C
A
Definition 3 (subcentral agent) Agent is called a agent in the -th cycle if it is not central and satisfies the following conditions for :
B
D
E
¯
F
¯
Figure 1. A GCP example
Now we introduce the enlarged neighborhood :
and send it to all its neighbors; Decide if it can become based on all the other received from its neighbors; If agent is , it will do the following: (a) If , changes the value of such
2. Compute 3. 4.
¼
that it can minimize the equation (6);
(b) If and there exists unsatisfied constraints associated with its represented variable, increases the Lagrange multipliers by one for all unsatisfied edges and sends them to associated neighbors.
This method is similar to the DB method [8]. According to definition 1, if and are central agents, and are not neighbors. Since only central agents can change the values of their variables, the behavior of each central agent cannot directly affect other central agents in a cycle. Thus, in each cycle, the behaviors of central agents are aggregated to a global behavior to optimize equation (6). When a central agent cannot optimize equation (6) by changing the value of its variable and there exist unsatisfied constraints associated with its variable, it means that the central agent gets stuck in a local minimum. Then, an update on the Lagrange multipliers associated with the unsatisfied constraints will help the agent escape from the local minimum. However, there exist some problems in this method. Consider the 3-colorable problem in Figure 1. Assume agents represent vertices , respectively and all vertices have the same color initially. We need three cycles to find a solution by the above strategy. Note that the behavior of the central agent cannot directly influences agent . If can change its color simultaneously, a solution can be found with two cycles. As motivated by this observation, we introduce the notion of enlarged neighborhood to improve the performance.
3.2 Enlarged Neighborhood and Subcentral Agents Definition 2 (frozen agent) Agent is frozen in the -th cycle iff that is central.
, is frozen; if , either or is frozen; .
¯ if
such
¼ ¾
¼
Thus, for each agent , we can redefine its local behaviors in the -th cycle as follows: 1. Exchange the value information of its represented variable to its neighbors; 2. Compute bors;
and send it to its enlarged neigh-
If agent is , it will perform the following behaviors ¼
from its enlarged neighbors 3. Receive all the agent in this cycle; and decide if it can become a
4.
to change its local environment:
- message (the content is that ) to all agents in . , change the value of its repre-
(a) Send a agent is
(b) If sented variables;
and there exist unsatisfied con(c) If straints associated with its represented variable, increase the Lagrange multipliers by one for all unsatisfied edges and send them to associated neighbors.
- messages and decide if it can become agent in this cycle; If agent is , change the value of its represented variable when .
5. Receive a 6.
3.3 Cutoff Strategy A potential problem of equations (3) and (4) lies in the unlimited increase in some Lagrange multipliers. This may make the multi-agent system much difficult to escape from a local optimum. Many methods have been proposed to keep relatively even multipliers for centralized systems [4, 5, 6]. A rapid restart strategy [2] was introduced to eliminate the heavy-tail behavior of a backtrack procedure for propositional satisfiability and constraint satisfaction problems. In order to make the multipliers relatively even in distributed GCP solving, we introduce a cutoff threshold for Lagrange multipliers. The theshold is a value of cycles that triggers the system to reset Lagrange multipliers to zero. We will apply this strategy to the global control schedule.
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
Table 1. A comparison between DDLM DDLM . Testset Miles
REG (mulsol) REG (zeroin)
Instance miles500 miles750 miles1000 mulsol.i.1 mulsol.i.2 mulsol.i.3 zeroin.i.1 zeroin.i.2 zeroin.i.3
and
DDLM 212 1,082 1,684 1,645 1,910 1,988 3,046 1,986 2,022
DDLM 233 1,153 1,722 1,738 1,997 2,089 3,294 2,049 2,028
Table 2. A comparison among DDLM DDLM , and DDLM . Instance queen6 6 queen7 7 queen8 8 queen9 9
DDLM 2,008 1,724 3,874 2,552 7,688 4,661 11,943 6,519
DDLM 2,860 1,055 15,564 3,404 9,386 5,232 10,089 7,129
5 Conclusion
,
DDLM 1,642 1,080 2,043 1,631 5,391 3,788 7,801 5,727
4 Experiments and Discussions We conducted our experiments on feasible benchmark GCP instances: Miles graphs, REG graphs, and Queen graphs [1]. We ran each instance 100 times with random initial assignments. Initially, we set Lagrange multipliers to zero. Table 1 shows the average numbers of cycles needed to find a solution by two DDLM methods with normal neighborhood (DDLM ) and enlarged neighborhood (DDLM ), respectively. As compared to DDLM , DDLM needs less cycles. In DDLM , each agent sends its to its enlarged neighborhood. If an agent becomes central in a certain cycle, it also sends to its enlarged neighborhood. This gives the agents in enlarged neighborhood a chance to consider their local behaviors in a more global level. Table 2 shows the experimental results of the DDLM methods with normal neighborhood (DDLM ), enlarged neighborhood (DDLM ), and cutoff value (DDLM ) on four Queen graph instances. In the experiment, we set the cutoff threshold to ¾¼¼. In Table 2, the first row corre-
sponding to each each instance shows the mean cycles and the second row shows the median cycles. We find that the performance in solving these instances depends on the initial assignment. This point is shown by the larger differences between the corresponding mean and median values. We further find that the cutoff strategy significantly reduces the number of cycles to find a solution in those instances. Since the strategy is to periodically reset Lagrange multipliers, it shows that keeping relatively even multipliers is a helpful for the performance of the DDLM methods.
We have presented a distributed discrete Lagrange multiplier (DDLM) method for solving distributed CSPs. According to this method, the local behaviors of agents are aggregated to find a global descent direction of an objective function corresponding to the CSP to be solved. Through experiments, we have validated the effectiveness of the DDLM method. We have improved its performance by introducing three techniques, i.e., normal neighborhood, enlarged neighborhood, and cutoff strategy.
References [1] http://mat.gsia.cmu.edu/COLORING03. [2] C. P. Gomes, B. Selman, N. Crato, and H. Kautz. Heavytailed phenomena in satisfiability and constraint satisfaction problems. Journal of Automated reasoning, 24:67–100, 2000. [3] J. Liu, H. Jing, and Y. Y. Tang. Multi-agent oriented constraint satisfaction. Artificial Intelligence, 138:101–144, 2002. [4] D. Schuurmans and F. Southey. Local search characteristics of incomplete SAT procedures. Artificial Intelligence, 132(2):121–150, 2001. [5] Y. Shang and B. W. Wah. A discrete lagrangian-based globalsearch method for solving satisfiability problems. Journal of global optimization, 12(1):61–99, 1998. [6] Z. Wu and B. W. Wah. An efficient global-search strategy in discrete lagrangian methods for solving hard satisfiability problems. In Proceedings of AAAI/IAAI 2000, pages 310– 315, 2000. [7] M. Yokoo, E. H. Durfee, T. Ishida, and K. Kuwabara. The distributed constraint satisfaction problem: Formalization and algorithms. IEEE Transaction on Knowledge and Data Engineering, 10(5):673–685, 1998. [8] M. Yokoo and K. Hirayama. Distributed breakout algorithm for solving distributed constraint satisfaction problems. In Proceedings of ICMAS96, pages 401–408, 1996.
Proceedings of the IEEE/WIC/ACM International Conference on Intelligent Agent Technology (IAT’04) 0-7695-2101-0/04 $ 20.00 IEEE
