On the complexity of indeterminate strings matching

Given two indeterminate equal-length strings p and t with a set of r characters per position in both strings, we obtain a determinate string p_w from p and a determinate string t_w from t by choosing one character per position. Then, we say that p and t match when p_w and t_w match for some choice of characters. We systematically study the complexity of string matching for indeterminate equal-length strings, for different notions of matching: parameterized matching, order-preserving matching, and Cartesian tree matching. We use n to denote the length of both strings and r the upper bound on the number of characters per position. First, we provide an algorithm for the Cartesian tree version that runs in O(nlognloglogn) time using O(n) space, for any constant r. Second, we establish NP-hardness of the order-preserving version for r=2, thus solving a question explicitly stated by Henriques et al. [CPM 2018], who showed hardness for r=3. Third, we establish NP-hardness of the parameterized version for r=2. As both parameterized and order-preserving indeterminate matching reduce to the standard determinate matching for r=1, this provides a complete classification for these three variants.