The Complexity of the Distributed Constraint Satisfaction Problem

TL;DR

This paper characterizes the computational complexity of the Distributed Constraint Satisfaction Problem (DCSP) on anonymous networks, showing that tractability depends on the invariance of the problem's template under symmetric polymorphisms.

Contribution

It provides a complete complexity classification for DCSP based on algebraic invariance properties, extending classical CSP results to distributed settings.

Findings

DCSP is polynomial-time solvable if and only if the template is invariant under symmetric polymorphisms.

If the template lacks this invariance, no finite-time algorithms exist for DCSP.

The neighborhood's iterated degree in the network is crucial for analyzing problem complexity.

Abstract

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.