Complexity of the homomorphism extension problem in the random case

TL;DR

This paper shows that for large random relational structures with at least one relation of arity two or more, the homomorphism extension problem is almost always NP-complete, indicating high computational complexity in typical cases.

Contribution

It establishes that the homomorphism extension problem is almost surely NP-complete for large random structures with certain relational properties, highlighting complexity in average-case scenarios.

Findings

Hom(A) is almost surely NP-complete for large random A with at least one relation of arity ≥ 2.

The result applies to a broad class of random relational structures.

Complexity persists in the typical case, not just in worst-case instances.

Abstract

We prove that if A is a large random relational structure with at least one relation of arity at least 2 then the problem EXT(A) is almost surely NP-complete.