What do Validiums prove?

Zero-knowledge proofs start with some solution to a problem that can be verified in polynomial time. The prover then generates curve points with a specific relationship. Generating those values without knowing the problem solution is not feasible. The verifier checks the relationship between the curve points.

For the specific case of a validium, what is the ‘solution’ here? Is it the list of leaf merkle hashes for a specific root? Is it the transactions? Is the validity of the transaction signatures a constraint of the solution?