We study the complexity of the Distributed Constraint Satisfaction Problem (DCSP) on a synchronous, anonymous network from a theoretical standpoint. In this setting, variables and constraints are controlled by agents which communicate with each other by sending messages through fixed communication channels. Our results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template's invariance under certain operations. Specifically, we show that DCSP($\Gamma$) is polynomial-time tractable if and only if $\Gamma$ is invariant under symmetric polymorphisms of all arities. Otherwise, there are no algorithms that solve DCSP($\Gamma$) in finite time. We also show that the same condition holds for the search variant of DCSP. Collaterally, our results unveil a feature of the processes' neighbourhood in a distributed network, its iterated degree, which plays a major role in the analysis. We explore this notion establishing a tight connection with the basic linear programming relaxation of a CSP.

翻译：我们从理论角度研究分布式限制满意度问题的复杂性。 在这种背景下,变量和制约因素由通过固定通信渠道发送信息而相互沟通的代理人控制。 我们的结果认可古典CSP的众所周知的事实,即固定板板计算问题的复杂性取决于某些操作中模板的变动。 具体地说,我们显示DCSP($\Gamma$)是多元时间的,如果而且只有在$\Gamma$是属于所有事物的对称多元形态的变异状态下才能进行。 否则,没有算法在有限的时间内解决DCSP($\Gamma$) 。 我们还表明,对于DCSP的搜索变异模式,我们的结果也持有同样的条件。 附带地说,我们的结果揭示了在分布式网络中流程邻里的一个特征,即其迭代程度,在分析中发挥了主要作用。 我们探索了这个概念,即与CSP的基本线性编程松动关系密切。