Decomposing constraint systems: equivalences and computational properties

Abstract

Distributed systems can often be modeled as a collection of distributed (system) variables whose values are constrained by a set of constraints. In distributed multi-agent systems, the set of variables occurring at a site (subsystem) is usually viewed as controllable by a local agent. This agent assigns values to the variables, and the aim is to provide distributed methods enabling a set of agents to come up with a global assignment (solution) that satisfies all the constraints. Alternatively, the system might be understood as a distributed database. Here, the focus is on ensuring consistency of the global system if local constraints (the distributed parts of the database) change. In this setting, the aim is to determine whether the existence of a global solution can be guaranteed. In other settings (e.g., P2P systems, sensor networks), the values of the variables might be completely out of control of the individual systems, and the constraints only characterize globally normal states or behavior of the system. In order to detect anomalies, one specifies distributed methods that can efficiently indicate violations of such constraints. The aim of this paper is to show that the following three main problems identified in these research areas are in fact identical: (i) the problem of ensuring that independent agents come up with a global solution; (ii) the problem of ensuring that global consistency is maintained if local constraint stores change; and (iii) the problem of ensuring that global violations can be detected by local nodes. This claim is made precise by developing a decomposition framework for distributed constraint systems and then extracting preservation properties that must satisfied in order to solve the above mentioned problems. Although satisfying the preservation properties seems to require different decomposition modes, our results demonstrate that in fact these decomposition properties are equivalent, thereby showing that the three main problems identified above are identical. We then show that the complexity of finding such decompositions is polynomially related to finding solutions for the original constraint system, which explains the popularity of decomposition applied to tractable constraint systems. Finally, we address the problem of finding optimal decompositions and show that even for tractable constraint systems, this problem is hard.