The Complexity of Network Satisfaction Problems for Symmetric Relation Algebras with a Flexible Atom

TL;DR

This paper classifies the computational complexity of network satisfaction problems for symmetric relation algebras with a flexible atom, showing they are either NP-complete or solvable in polynomial time.

Contribution

It provides a complete classification for symmetric, flexible-atom relation algebras, reducing the problem to integral cases and applying universal algebra and Ramsey theory techniques.

Findings

The problem is NP-complete or in P for the specified class.

Reduction to integral relation algebras simplifies the classification.

Use of universal algebra and Ramsey theory to analyze complexity.

Abstract

Robin Hirsch posed in 1996 the 'Really Big Complexity Problem': classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a flexible atom; in this case, the problem is NP-complete or in P. The classification task can be reduced to the case where A is integral. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Ne\v{s}et\v{r}il and R\"odl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.