Subsequences With Gap Constraints: Complexity Bounds for Matching and Analysis Problems

We consider subsequences with gap constraints, i.e., length-k subsequences p that can be embedded into a string w such that the induced gaps (i.e., the factors of w between the positions to which p is mapped to) satisfy given gap constraints gc = (C1, C2, ..., Ck-1); we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds Ci = (L^-i, L^+i) ∈ mathbbN2 and/or regular languages Ci ∈ REG, we prove tight (conditional on the orthogonal vectors (OV) hypothesis) complexity bounds for checking whether a given p is a gc-subsequence of a string w. We also consider the whole set of all gc-subsequences of a string, and investigate the complexity of the universality, equivalence and containment problems for these sets of gc-subsequences.