Openness in Asynchronous Constraint Satisfaction Algorithms Marius-C˘ alin Silaghi ∗ Florida Institute of Technology (FIT) Melbourne, Florida 32901-6988, USA msilaghi@cs.fit.edu Boi Faltings Swiss Federal Institute of Technology (EPFL) CH-1015 Lausanne, Switzerland Boi.Faltings@epfl.ch Abstract Constraint satisfaction occurs in many practical applica- tions. Recently, there has been increasing interest in dis- tributed constraint satisfaction (DisCSP), where variables and constraints are distributed among several agents, and algorithms for finding solutions through asynchronous ex- change of messages among agents. An important reason for distributed problem solving is openness: allowing the problem to define itself dynam- ically through the combination of agents that participate in it. We investigate openness in complete asynchronous search algorithms for DisCSP, in particular the problem of agents joining and leaving a search process in progress without destroying consistency for the other agents, and give complete search algorithms that satisfy this property. 1 Introduction Constraint satisfaction has been applied with great suc- cess to many important practical applications. Increas- ingly, it also occurs in distributed settings with variables and constraints defined by different agents. There has thus been significant interest in the distributed constraint satis- faction (DisCSP [29]) problem with asynchronous, dis- tributed algorithms for solving it. A major characteristic of distributed systems is their openness: the combination of agents making up the sys- tem is not built into the algorithms, and may not even be known when the algorithm is running, and agents may * This work was performed while the first author was working at EPFL, supported by the Swiss National Science Foundation project number 21-52462.97. even be joining and leaving the system while an algorithm is active. Algorithms for DisCSP have focussed on asyn- chronous execution, but have not considered openness to a great extent. In particular, asynchronous search algorithms for DisCSP assume that the set of agents, variables and con- straints is known at the beginning of the search and fixed throughout. In this paper, we show how to create the pos- sibility for agents to join and leave an ongoing search pro- cess without halting it, and reusing the results of the on- going search as much as possible. In general, constraint satisfaction algorithms can be easily adapted to allow additions of constraints during search. Similarly, it turns out to be straightforward to al- low new agents to introduce their variables and constraints in DisCSP search without need to restart an ongoing asyn- chronous search process. It is however considerably more complex to allow an agent to leave and remove its variables and constraints, since this may make valid assignments which had earlier been discarded. For the centralized case, Bessiere ([3]) has shown how to dynamically put back values elimi- nated by consistency techniques, so that search can restart with the variable or constraint that has been removed rather than from the beginning. Verfaille and Schiex ([6]) has shown techniques for adapting complete solutions to changes in the constraint system. Both techniques only address centralized search algorithms, and it is not clear how they could be generalized to an asynchronous case. A central innovation we present in this paper is a relax- ation technique for asynchronous, distributed search al- gorithms that allows us to reactive the right parts of the search space when agents and their associated variables and constraints leave an ongoing asynchronous search. 1