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

TL;DR

This paper investigates the computational complexity of identifying and analyzing subsequences with specific gap constraints, providing tight bounds and exploring related decision problems under various conditions.

Contribution

It establishes tight complexity bounds for gc-subsequence matching and analyzes problems like universality, equivalence, and containment for these subsequences.

Findings

Conditional tight complexity bounds for gc-subsequence matching

Complexity analysis of universality, equivalence, and containment problems

Results depend on gap constraint definitions and the OV hypothesis

Abstract

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 $gc = (C_1, C_2, ..., C_{k-1})$; we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds $C_i = (L^-_i, L^+_i) \in \mathbb{N}^2$ and/or regular languages $C_i \in REG$, 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.