# Centralized Task Allocation and Alignment Based on Constraint Table and Alignment Rules
## 1 Introduction
First, the groups of tasks are made based on coupled constraints, called activities. Dependency matrices and temporal constraint matrices are made for each activity to obtain a solution applying constraints. Then, each agent decides optimistic or pessimistic strategies for bidding on each task. For bidding, each agent counts violation iterations, number of bidding alone to task, and number of bidding of any task. This process is also time-consuming. There is another way to allocate and align tasks from a different point of view, game theory [15]. In this method, the individual agent has freedom for its actions and the agent decides its action by sampling from its possible action set stochastically. When the agent violates the constraints for collective action, penalties that are larger than other rewards are given to it. These penalties are added to the score according to time-discounted reward scheme, so each agent is constrained to obey the constraints. This method solved a minor problem of CCBBA as setting the temporal constraints by just the relationship, such as 'before' or 'after' etc., so there is no need to set permitted time, such as 'do in 30 min after task 1'. While CBBA and CCBBA did not permit the task to be allocated to multi-agents, task allocation based on game theory permits collective actions for one task. In this case, duration time of the task decreases because several agents work together. However, this method results in acceptable and stable solutions only when the probability distributions of actions in action set for each agent are set based on large experiments. It means that a lot of experiments for several cases are needed to obtain the suitable probability distributions. Additionally, computation time is reduced compared to CCBBA, but it is still slow so the update interval of task allocation and alignment results is long. According to the results in [15], 18.086 s (mean time) to obtain reasonable central task allocation solution with six agents and 27 tasks. In this paper, we consider an algorithm for coupled-constraint task allocation and alignment problems using a constraint table and alignment rules. This is based on CBBA and CCBBA, but a new method is proposed alternating some time-consuming procedures, so local optimum solutions are obtained fast enough. For the task allocation, two scoring schemes are proposed; modified time-discounted scoring scheme and accumulated distance scoring scheme. Additionally, a spatial constraint in CCBBA is considered, called MUTually EXclusive (MUTEX). In addition, a new spatial constraint, called local MUTEX, is considered for essential cases; cooperative work, and etc. For classification, MUTEX in CCBBA is called global MUTEX. Unilateral dependency in CCBBA is considered by temporal constraints instead. For the task alignment, some temporal constraints in CCBBA are considered afterwards, called simultaneous. Temporal constraints 'before' and 'between' are replaced to 'after'. Additionally, constraints 'during' is separated to two constraints;'start during' and 'end during'. Constraint 'not during' in CCBBA is not used. These temporal constraints are considered using the constraint table which is made when alignment of a task is completed. To apply these constraints and scoring schemes to the task allocation and alignment, the algorithm called centralized Task Allocation and alignment based on Constraint Table and Alignment Rules (TACTAR) is proposed.