Hardness of Network Satisfaction for Relation Algebras with Normal Representations

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; the problem is in this case NP-complete or in P. 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 Nešetřil and Rdl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.

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

“…Example 6.8 (Hardness of representable relation algebra #17, see Bodirsky & Knäuer, 2020). Let N be the normal representation of the algebra #17 mentioned in Example 3.9.…”

Network Satisfaction for Symmetric Relation Algebras with a Flexible Atom

“…The following shows how to apply our hardness result to a concrete A ∈ RRA. Example 37 (Hardness of relation algebra #17, see Bodirsky and Knäuer 2020a)).…”

Solving Infinite-Domain CSPs Using the Patchwork Property

“…Moreover, Hirsch [42] proposed studying the computational complexity of CSPs for relation algebras, with obvious applications in AI. Inspired by this research programme, Bodirsky and Knäuer [19] recently identified sufficient conditions for homogeneity of relation algebras. Their results provide further examples of CSPs that are covered by Theorem 6.…”