A note on the product homomorphism problem and CQ-definability

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 × · · · × An to B. This problem is clearly solvable in non-deterministic exponential time. It follows from results in [1] that the problem is NExpTimecomplete. 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). More precisely, we obtain: