A note on the product homomorphism problem and CQ-definability

TL;DR

This paper proves the NExpTime-completeness of the product homomorphism problem (PHP), including for directed graphs and structures with bounded arity, and explores implications for CQ-definability.

Contribution

It provides a self-contained proof of PHP's complexity and extends the hardness results to specific classes of structures, with applications to CQ-definability.

Findings

PHP is NExpTime-complete for directed graphs.

PHP remains hard for structures with bounded arity and domain size.

The results have implications for CQ-definability complexity.

Abstract

The product homomorphism problem (PHP) takes as input a finite collection of relational structures A1, ..., An and another relational structure B, all over the same schema, and asks whether there is a homomorphism from the direct product A1 x ... x An to B. This problem is clearly solvable in non-deterministic exponential time. It follows from results in [1] that the problem is NExpTime-complete. The proof, based on a reduction from an exponential tiling problem, uses structures of bounded domain size but with relations of unbounded arity. In this note, we provide a self-contained proof of NExpTime-hardness of PHP, and we show that it holds already for directed graphs, as well as for structures of bounded arity with a bounded domain size (but without a bound on the number of relations). We also present an application to the CQ-definability problem (also known as the PP-definability problem). [1] Ross Willard. Testing expressibility is hard. In David Cohen, editor, CP, volume 6308 of Lecture Notes in Computer Science, pages 9-23. Springer, 2010.