The complexity of Shortest Common Supersequence for inputs with no identical consecutive letters

TL;DR

This paper investigates the computational complexity of two variants of the Shortest Common Supersequence problem when inputs have no consecutive identical letters, proving NP-completeness for these variants with alphabet size at least 3.

Contribution

It establishes that the variants S and MSCS remain NP-complete for alphabet sizes of three or more, extending previous results to a broader class of inputs.

Findings

Both S and MSCS variants are NP-complete for |S| .

NP-completeness holds even when inputs have no consecutive identical letters.

The results extend known complexity bounds to smaller alphabet sizes.

Abstract

The Shortest Common Supersequence problem (SCS for short) consists in finding a shortest common supersequence of a finite set of words on a fixed alphabet Sigma. It is well-known that its decision version denoted [SR8] in [Garey and Johnson] is NP-complete. Many variants have been studied in the literature. In this paper we settle the complexity of two such variants of SCS where inputs do not contain identical consecutive letters. We prove that those variants denoted \varphi SCS and MSCS both have a decision version which remains NP-complete when |\Sigma| is at least 3. Note that it was known for MSCS when |\Sigma| is at least 4 [Fleisher and Woeginger] and we discuss how [Darte] states a similar result for |\Sigma| at least 3.