Approximation and Fixed Parameter Algorithms for the Approximate Cover Problem

Amir et al. (CPM 2017) introduce the approximate string cover problem (ACP) motivated by applications including molecular biology, coding, automata theory, formal language theory and combinatorics. A cover of a string T is a string C for which every letter of T lies within some occurrence of C. The input of the ACP consists of a string T and the goal is to find a string C of length less than the length of T that covers a string T ', which is as close to T as possible (under some predefined distance). Amir et al. study this problem for the Hamming distance and show that it is NP-hard. In this paper we continue the work of Amir et al. and show the following results for the cover length relaxation of the ACP. After observing that the NP-hardness proof by Amir et al. (CPM 2017, TCS 2019) suffers from several lapses, we propose an amendment to the proof. We then introduce an approximation algorithm for a variant of the ACP, in which we aim to maximize the length of the input string minus the distance to the string covered by the approximate cover returned by the algorithm. This problem is naturally as hard as the ACP. We prove an asymptotic approximation ratio of O(root|T|), where |T| is the size of the input string. Finally, we present an FPT algorithm with respect to the alphabet size and the size of the cover based on a dynamic programming framework.