More on the complexity of common superstring and supersequence problems

Abstract

The Shortest Common Superstring (SCSt) problem is the decision version of the problem to find, for a finite set of strings L over an alphabet Σ and an integer k ϵ N , a string S of length ⩽ k over Σ such that each string in L is a substring of S. Analogously, the Shortest Common Supersequence (SCSe) problem is the decision version of the problem to find, for a finite set of strings L over an alphabet Σ and an integer k ϵ N , a string S of length ⩽ k over Σ such that each string in L is a subsequence of S. Our main results are: • SCSt is NP-complete if the given strings have length 3 and the maximal orbit size is 8 (the orbit size of a character is the number of its occurrences in the strings in L). This partially solves a problem of Timkovskii (1990). • SCSt is NP-complete over the alphabet {0, 1} even if each given string contains exactly three ones. • SCSe over the alphabet {0, 1} is NP-complete even if the given strings all have the same length and each string contains exactly two ones. Moreover, we introduce cyclic and permutation variants of SCSt and SCSe, namely Cyclic-SCSt, Cyclic-SCSe, Permutation-SCSt and Permutation-SCSe. The Permutation-SCSt (resp. Cyclic-SCSt) problem is the decision version of the problem to find, for a finite set of strings L over an alphabet Σ and an integer k ϵ N , a string S of length ⩽ k over Σ such that there exists a (cyclic) permutation of each string in L that is a substring of S. Permutation-SCSe and Cyclic-SCSe are defined analogously. The main results are: • Cyclic-SCSt and Cyclic-SCSe are NP-complete for strings of length 3 and polynomial-time-solvable for string of length 2. • Cyclic-SCSt and Cyclic-SCSe are NP-complete for an alphabet of size 2. • Permutation-SCSt is NP-complete if the given strings have length 3 and the maximal orbitsize is 8. • Permutation-SCSt is NP-complete for an alphabet of size 3 and polynomial-time-solvable for an alphabet of size 2.